Algebra: Introductory and Intermediate, Fourth Edition

Fourth Edition Algebra: Introductory and Intermediate Richard N. Aufmann Palomar College, California Vernon C. Barke...

Author: Richard N. Aufmann | Vernon C. Barker | Joanne Lockwood

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Fourth Edition

Algebra: Introductory and Intermediate

Richard N. Aufmann Palomar College, California

Vernon C. Barker Palomar College, California

Joanne S. Lockwood New Hampshire Community Technical College

Houghton Mifflin Company Boston

New York

Associate Editor: Melissa Parkin Assistant Editor: Noel Kamm Senior Project Editor: Tamela Ambush Editorial Assistant: Sage Anderson Manufacturing Buyer: Florence Cadran Executive Marketing Manager: Brenda Bravener-Greville Senior Marketing Manager: Jennifer Jones Cover photo: © Karim Rashid Photo Credits: CHAPTER 1: p. 1, Gary Conner/PhotoEdit; p. 3, Tony Freeman/PhotoEdit; p. 15, Paula Bronstein/ Getty Images; p. 36, AP/Wide World Photos; p. 53, Image Bank/Getty Images. CHAPTER 2: p. 71, Michael Newman/PhotoEdit; p. 77, ©Terres Du Sud/CORBIS; p. 89, Martin Fox/Index Stock Imagery; p. 89, Bill Aron/PhotoEdit; p. 90, ©Lawrence Manning/CORBIS; p. 91, Davis Barber/ PhotoEdit; p. 111, ©Steve Prezant/CORBIS; p. 121, ©Renee Comet/PictureArts Corp/CORBIS; p. 122, Photodisc; p. 135, ©Richard Cummins/CORBIS; p. 153, Alan Oddie/PhotoEdit. CHAPTER 3: p. 159, ©Kevin Fleming/CORBIS; p. 176, ©2002 Photodisc; p. 177, ©CORBIS; p. 190, ©Topham/Syndicated Features Limited/The Image Works; p. 192, ©Roger Ressmeyer/ CORBIS; p. 192, ©CORBIS; p. 201, ©Kevin R. Morris/CORBIS. CHAPTER 4: p. 217, Jeff Greenberg/PhotoEdit; p. 223, ©Craig Tuttle/CORBIS; p. 229, Ulrike Welsh, PhotoEdit; p. 239, Robert W. Ginn/PhotoEdit; p. 260, Eric Fowke/PhotoEdit; p. 270, ©CORBIS; p. 271, ©David Keaton/CORBIS; p. 283, The Granger Collection, New York. CHAPTER 5: p. 295, David R. Stoecklein/CORBIS; p. 325, Jose Carillo/PhotoEdit; p. 326, ©CORBIS; p. 327, Michael Newman/ PhotoEdit; p. 333, AP/World Wide Photos; p. 335, Michael Newman/PhotoEdit; p. 340, Susan Van Etten/PhotoEdit. CHAPTER 6: p. 343, AP/World Wide Photos; p. 355, Stocktrek/CORBIS; p. 356, ©NASA/JPL Handout/Reuters/CORBIS; p. 356, John Neubauer/PhotoEdit; p. 387, ©Duomo/ CORBIS; p. 388, ©Roger Ressmeyer/CORBIS; p. 394, Susan Van Etten/PhotoEdit. CHAPTER 7: p. 399, Digital Vision/Getty Images; p. 439, ©Pierre Ducharme/Reuters/CORBIS; p. 439, AP/Wide World Photos; p. 441, Bill Aron/PhotoEdit; p. 446, ©CORBIS. CHAPTER 8: p. 451, ©Jeff Henry/ Peter Arnold, Inc.; p. 478, Clayton Sharrard/PhotoEdit; p. 482, ©CORBIS; p. 482, ©Alinari Archives/CORBIS; p. 493, Tom Carter/PhotoEdit; p. 493, David Young-Wolff/PhotoEdit; p. 494, ©Sheldan Collins/CORBIS; p. 494, ©Galen Rowell/CORBIS; p. 495, Billy E. Barnes/PhotoEdit; p. 496, ©Lee Cohen/CORBIS; p. 499, Tony Freeman/PhotoEdit; p. 512, ©CORBIS. CHAPTER 9: p. 515, Benjamin Shearn/TAXI/Getty Images; p. 539, ©Bettmann/CORBIS; p. 541, Sandor Szabo/ EPA/Landov; CHAPTER 10: p. 561, David Young-Wolff/PhotoEdit; p. 576, Photex/CORBIS; p. 595, Bill Aron/PhotoEdit; p. 596, ©Reuters/CORBIS. CHAPTER 11: p. 607, Tim Boyle/Getty Images; p. 608, Lon C. Diehl/PhotoEdit; p. 616, ©Jim Craigmyle/CORBIS; p. 623, Rich Clarkson/ Getty Images; p. 624, Vic Bider/PhotoEdit; p. 627, ©Nick Wheeler/CORBIS; p. 633, ©Jose Fuste Raga/CORBIS; p. 637, ©Joel W. Rogers/CORBIS; p. 638, Robert Brenner/PhotoEdit; p. 649, Michael Newman/PhotoEdit; p. 658, ©Kim Sayer/CORBIS; CHAPTER 12: p. 659, Rudi Von Briel/ PhotoEdit; p. 672, The Granger Collection, New York; p. 689, Express Newspaper/Getty Images; p. 690, ©Richard T. Nowitz/CORBIS; p. 690, Courtesy of the Edgar Fahs Smith Image Collection/ University of Pennsylvania Library, Philadelphia, PA 19104-6206; p. 691, ©Bettman/CORBIS; p. 693, Mark Harmel/Getty Images; p. 694, Myrleen Fergusun Cate//PhotoEdit; p. 695, Mike Johnson-www.earthwindow.com; p. 696, ©Roger Ressmeyer/CORBIS; p. 699, David Young-Wolff/ PhotoEdit; p. 704, ©Macduff Everton/CORBIS; p. 708, Frank Siteman/PhotoEdit.

Copyright © 2007 by Houghton Mifflin Company. All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system without the prior written permission of Houghton Mifflin Company unless such copying is expressly permitted by federal copyright law. Address inquiries to College Permissions, Houghton Mifflin Company, 222 Berkeley Street, Boston, MA 02116-3764. Printed in the U.S.A. Library of Congress Control Number: 2005936314 Instructor’s Annotated Edition: ISBN 13: 978-0-618-61131-7 ISBN 10: 0-618-61131-2 For orders, use student text ISBNs: ISBN 13: 978-0-618-60953-6 ISBN 10: 0-618-60953-9 123456789-WC-10 09 08 07 06

Contents

Preface xi AIM for Success xxiii

1

Real Numbers and Variable Expressions 1 Prep Test 2

Section 1.1

Integers 3 Objective Objective Objective Objective Objective

Section 1.2

To To To To To

use inequality symbols with integers 3 find the additive inverse and absolute value of a number 4 add or subtract integers 5 multiply or divide integers 7 solve application problems 10

Rational and Irrational Numbers 17 Objective Objective Objective Objective Objective Objective Objective

Section 1.3

A B C D E

A B C D E F G

To To To To To To To

write a rational number as a decimal 17 convert among percents, fractions, and decimals 18 add or subtract rational numbers 19 multiply or divide rational numbers 20 evaluate exponential expressions 22 simplify numerical radical expressions 24 solve application problems 26

The Order of Operations Agreement 33 Objective A To use the Order of Operations Agreement to simplify expressions 33

Section 1.4

Variable Expressions 37 Objective A To evaluate a variable expression 37 Objective B To simplify a variable expression using the Properties of Addition 38 Objective C To simplify a variable expression using the Properties of Multiplication 40 Objective D To simplify a variable expression using the Distributive Property 42 Objective E To translate a verbal expression into a variable expression 44

Section 1.5

Sets 55 Objective A To write a set using the roster method 55 Objective B To write a set using set-builder notation 56 Objective C To graph an inequality on the number line 57

Focus on Problem Solving: Inductive Reasoning 61 • Projects and Group Activities: Calculators 62 • Chapter 1 Summary 63 • Chapter 1 Review Exercises 67 • Chapter 1 Test 69

2

First-Degree Equations and Inequalities 71 Prep Test 72

Section 2.1

Introduction to Equations 73 Objective A To determine whether a given number is a solution of an equation 73 Objective B To solve an equation of the form x a b 74

iii

Objective C To solve an equation of the form ax b 75 Objective D To solve application problems using the basic percent equation 77 Objective E To solve uniform motion problems 81

Section 2.2

General Equations 92 Objective Objective Objective Objective

Section 2.3

A B C D

To To To To

solve solve solve solve

an equation of the form ax b c 92 an equation of the form ax b cx d 95 an equation containing parentheses 97 application problems using formulas 99

Translating Sentences into Equations 106 Objective A To solve integer problems 106 Objective B To translate a sentence into an equation and solve 108

Section 2.4

Mixture and Uniform Motion Problems 113 Objective A To solve value mixture problems 113 Objective B To solve percent mixture problems 115 Objective C To solve uniform motion problems 117

Section 2.5

First-Degree Inequalities 125 Objective A To solve an inequality in one variable 125 Objective B To solve a compound inequality 128 Objective C To solve application problems 130

Section 2.6

Absolute Value Equations and Inequalities 137 Objective A To solve an absolute value equation 137 Objective B To solve an absolute value inequality 139 Objective C To solve application problems 141

Focus on Problem Solving: Trial-and-Error Approach to Problem Solving 147 • Projects and Group Activities: Water Displacement 148 • Chapter 2 Summary 149 • Chapter 2 Review Exercises 152 • Chapter 2 Test 155 • Cumulative Review Exercises 157

3

Geometry 159 Prep Test 160

Section 3.1

Introduction to Geometry 161 Objective A To solve problems involving lines and angles 161 Objective B To solve problems involving angles formed by intersecting lines 166 Objective C To solve problems involving the angles of a triangle 169

Plane Geometric Figures 177 Objective A To solve problems involving the perimeter of a geometric figure 177 Objective B To solve problems involving the area of a geometric figure 182

Section 3.3

Solids 195 Objective A To solve problems involving the volume of a solid 195 Objective B To solve problems involving the surface area of a solid 198

Focus on Problem Solving: More on the Trial-and-Error Approach to Problem Solving 205 • Projects and Group Activities: Investigating Perimeter 206 • Symmetry 207 • Chapter 3 Summary 207 • Chapter 3 Review Exercises 211 • Chapter 3 Test 213 • Cumulative Review Exercises 215

Copyright © Houghton Mifflin Company. All rights reserved.

Section 3.2

Contents

4

v

Linear Functions and Inequalities in Two Variables 217 Prep Test 218

Section 4.1

The Rectangular Coordinate System 219 Objective A To graph points in a rectangular coordinate system 219 Objective B To determine ordered-pair solutions of an equation in two variables 221 Objective C To graph a scatter diagram 223

Section 4.2

Introduction to Functions 229 Objective A To evaluate a function 229

Section 4.3

Linear Functions 241 Objective Objective Objective Objective

Section 4.4

A B C D

To To To To

graph a linear function 241 graph an equation of the form Ax By C 243 find the x- and y-intercepts of a straight line 246 solve application problems 248

Slope of a Straight Line 253 Objective A To find the slope of a line given two points 253 Objective B To graph a line given a point and the slope 257

Section 4.5

Finding Equations of Lines 264 Objective A To find the equation of a line given a point and the slope 264 Objective B To find the equation of a line given two points 265 Objective C To solve application problems 267

Section 4.6

Parallel and Perpendicular Lines 273 Objective A To find parallel and perpendicular lines 273

Section 4.7

Inequalities in Two Variables 279 Objective A To graph the solution set of an inequality in two variables 279

Focus on Problem Solving: Find a Pattern 283 • Projects and Group Activities: Introduction to Graphing Calculators 284 • Chapter 4 Summary 285 • Chapter 4 Review Exercises 288 • Chapter 4 Test 291 • Cumulative Review Exercises 293

5

Systems of Linear Equations and Inequalities 295


Prep Test 296 Section 5.1

Solving Systems of Linear Equations by Graphing and by the Substitution Method 297 Objective A To solve a system of linear equations by graphing 297 Objective B To solve a system of linear equations by the substitution method 300 Objective C To solve investment problems 303

Section 5.2

Solving Systems of Linear Equations by the Addition Method 309 Objective A To solve a system of two linear equations in two variables by the addition method 309 Objective B To solve a system of three linear equations in three variables by the addition method 312

Section 5.3

Application Problems 321 Objective A To solve rate-of-wind or rate-of-current problems 321 Objective B To solve application problems 322

Contents

Section 5.4

Solving Systems of Linear Inequalities 329 Objective A To graph the solution set of a system of linear inequalities 329

Section 5.5

Solving Systems of Equations by Using Determinants

(Available only online at this

textbook’s website at math.college.hmco.com/students. Under Developmental Mathematics, select course area, then select textbook.)

Focus on Problem Solving: Solve an Easier Problem 333 • Projects and Group Activities: Using a Graphing Calculator to Solve a System of Equations 333 • Chapter 5 Summary 335 • Chapter 5 Review Exercises 337 • Chapter 5 Test 339 • Cumulative Review Exercises 341

6

Polynomials 343 Prep Test 344

Section 6.1

Exponential Expressions 345 Objective A To multiply monomials 345 Objective B To divide monomials and simplify expressions with negative exponents 347 Objective C To write a number using scientific notation 351 Objective D To solve application problems 352

Section 6.2

Introduction to Polynomial Functions 357 Objective A To evaluate polynomial functions 357 Objective B To add or subtract polynomials 360

Section 6.3

Multiplication of Polynomials 365 Objective Objective Objective Objective

Section 6.4

A B C D

To To To To

multiply a polynomial by a monomial 365 multiply two polynomials 366 multiply polynomials that have special products 368 solve application problems 369

Division of Polynomials 375 Objective Objective Objective Objective

A B C D

To To To To

divide a polynomial by a monomial 375 divide polynomials 376 divide polynomials by using synthetic division 378 evaluate a polynomial function using synthetic division 380

Focus on Problem Solving: Dimensional Analysis 386 • Projects and Group Activities: Astronomical Distances and Scientific Notation 388 • Chapter 6 Summary 389 • Chapter 6 Review Exercises 392 • Chapter 6 Test 395 • Cumulative Review Exercises 397

7

Factoring 399 Prep Test 400

Section 7.1

Common Factors 401 Objective A To factor a monomial from a polynomial 401 Objective B To factor by grouping 403

Section 7.2

Factoring Polynomials of the Form x 2 bx c Objective A To factor a trinomial of the form Objective B To factor completely 409

Section 7.3

x2

407

bx c 407

Factoring Polynomials of the Form ax 2 bx c

415

Objective A To factor a trinomial of the form ax 2 bx c by using trial factors 415


vi

Contents

vii

Objective B To factor a trinomial of the form ax 2 bx c by grouping 417

Section 7.4

Special Factoring 423 Objective A To factor the difference of two perfect squares or a perfect-square trinomial 423 Objective B To factor the sum or difference of two perfect cubes 425 Objective C To factor a trinomial that is quadratic in form 426 Objective D To factor completely 427

Section 7.5

Solving Equations 433 Objective A To solve equations by factoring 433 Objective B To solve application problems 435

Focus on Problem Solving: Making a Table 441 • Projects and Group Activities: Exploring Integers 442 • Chapter 7 Summary 442 • Chapter 7 Review Exercises 445 • Chapter 7 Test 447 • Cumulative Review Exercises 449

8

Rational Expressions 451 Prep Test 452

Section 8.1

Multiplication and Division of Rational Expressions 453 Objective A To simplify a rational expression 453 Objective B To multiply rational expressions 454 Objective C To divide rational expressions 456

Section 8.2

Addition and Subtraction of Rational Expressions 461 Objective A To rewrite rational expressions in terms of a common denominator 461 Objective B To add or subtract rational expressions 463

Section 8.3

Complex Fractions 469 Objective A To simplify a complex fraction 469

Section 8.4

Solving Equations Containing Fractions 473 Objective A To solve an equation containing fractions 473

Section 8.5

Ratio and Proportion 477 Objective A To solve a proportion 477 Objective B To solve application problems 478 Objective C To solve problems involving similar triangles 478


Section 8.6

Literal Equations 485 Objective A To solve a literal equation for one of the variables 485

Section 8.7

Application Problems 489 Objective A To solve work problems 489 Objective B To use rational expressions to solve uniform motion problems 491

Section 8.8

Variation 497 Objective A To solve variation problems 497

Focus on Problem Solving: Implication 503 • Projects and Group Activities: Intensity of Illumination 504 • Chapter 8 Summary 505 • Chapter 8 Review Exercises 509 • Chapter 8 Test 511 • Cumulative Review Exercises 513

Contents

9

Exponents and Radicals 515 Prep Test 516

Section 9.1

Rational Exponents and Radical Expressions 517 Objective A To simplify expressions with rational exponents 517 Objective B To write exponential expressions as radical expressions and to write radical expressions as exponential expressions 519 Objective C To simplify radical expressions that are roots of perfect powers 521

Section 9.2

Operations on Radical Expressions 527 Objective Objective Objective Objective

Section 9.3

A B C D

To To To To

simplify radical expressions 527 add or subtract radical expressions 528 multiply radical expressions 529 divide radical expressions 531

Solving Equations Containing Radical Expressions 537 Objective A To solve a radical equation 537 Objective B To solve application problems 539

Section 9.4

Complex Numbers 543 Objective Objective Objective Objective

A B C D

To To To To

simplify a complex number 543 add or subtract complex numbers 544 multiply complex numbers 545 divide complex numbers 548

Focus on Problem Solving: Polya’s Four-Step Process 551 • Projects and Group Activities: Solving Radical Equations with a Graphing Calculator 552 • Chapter 9 Summary 553 • Chapter 9 Review Exercises 555 • Chapter 9 Test 557 • Cumulative Review Exercises 559

10

Quadratic Equations 561 Prep Test 562

Section 10.1

Solving Quadratic Equations by Factoring or by Taking Square Roots 563 Objective A To solve a quadratic equation by factoring 563 Objective B To write a quadratic equation given its solutions 564 Objective C To solve a quadratic equation by taking square roots 565

Section 10.2

Solving Quadratic Equations by Completing the Square 571 Objective A To solve a quadratic equation by completing the square 571

Section 10.3

Solving Quadratic Equations by Using the Quadratic Formula 577 Objective A To solve a quadratic equation by using the quadratic formula 577

Section 10.4

Solving Equations That Are Reducible to Quadratic Equations 583 Objective A To solve an equation that is quadratic in form 583 Objective B To solve a radical equation that is reducible to a quadratic equation 584 Objective C To solve a fractional equation that is reducible to a quadratic equation 586

Section 10.5

Quadratic Inequalities and Rational Inequalities 589 Objective A To solve a nonlinear inequality 589

Section 10.6

Applications of Quadratic Equations 593 Objective A To solve application problems 593


viii

Contents

ix

Focus on Problem Solving: Using a Variety of Problem-Solving Techniques 597 • Projects and Group Activities: Using a Graphing Calculator to Solve a Quadratic Equation 597 • Chapter 10 Summary 598 • Chapter 10 Review Exercises 601 • Chapter 10 Test 603 • Cumulative Review Exercises 605

11

Functions and Relations 607 Prep Test 608

Section 11.1

Properties of Quadratic Functions 609 Objective Objective Objective Objective

Section 11.2

A B C D

To To To To

graph a quadratic function 609 find the x-intercepts of a parabola 612 find the minimum or maximum of a quadratic function 615 solve application problems 616

Graphs of Functions 625 Objective A To graph functions 625

Section 11.3

Algebra of Functions 631 Objective A To perform operations on functions 631 Objective B To find the composition of two functions 633

Section 11.4

One-to-One and Inverse Functions 639 Objective A To determine whether a function is one-to-one 639 Objective B To find the inverse of a function 640

Section 11.5

Conic Sections

(Available only online at this textbook’s website at math.college.hmco.com/students.

Under Developmental Mathematics, select course area, then select textbook.)

Focus on Problem Solving: Algebraic Manipulation and Graphing Techniques 647 • Projects and Group Activities: Finding the Maximum or Minimum of a Function Using a Graphing Calculator 648 • Business Applications of Maximum and Minimum Values of Quadratic Functions 648 • Chapter 11 Summary 650 • Chapter 11 Review Exercises 653 • Chapter 11 Test 655 • Cumulative Review Exercises 657

12

Exponential and Logarithmic Functions 659 Prep Test 660

Section 12.1

Exponential Functions 661


Objective A To evaluate an exponential function 661 Objective B To graph an exponential function 663

Section 12.2

Introduction to Logarithms 668 Objective A To find the logarithm of a number 668 Objective B To use the Properties of Logarithms to simplify expressions containing logarithms 671 Objective C To use the Change-of-Base Formula 674

Section 12.3

Graphs of Logarithmic Functions 679 Objective A To graph a logarithmic function 679

Section 12.4

Solving Exponential and Logarithmic Equations 683 Objective A To solve an exponential equation 683 Objective B To solve a logarithmic equation 685

Section 12.5

Applications of Exponential and Logarithmic Functions 689 Objective A To solve application problems 689

Contents

Focus on Problem Solving: Proof by Contradiction 697 • Projects and Group Activities: Solving Exponential and Logarithmic Equations Using a Graphing Calculator 698 • Credit Reports and FICO® Scores 699 • Chapter 12 Summary 700 • Chapter 12 Review Exercises 702 • Chapter 12 Test 705 • Cumulative Review Exercises 707

Final Exam

R Section R.1

709

Review of Introductory Algebra Topics 713 Variable Expressions 715 Objective A To evaluate a variable expression 715 Objective B To simplify a variable expression 716

Section R.2

Equations and Inequalities 721 Objective A To solve a first-degree equation in one variable 721 Objective B To solve an inequality in one variable 723

Section R.3

Linear Equations in Two Variables 729 Objective Objective Objective Objective

Section R.4

A B C D

To To To To

graph points in a rectangular coordinate system 729 graph a linear equation in two variables 730 evaluate a function 737 find the equation of a line 738

Polynomials 743 Objective Objective Objective Objective Objective

A B C D E

To To To To To

multiply and divide monomials 743 add and subtract polynomials 745 multiply polynomials 746 divide polynomials 748 factor polynomials of the form ax 2 bx c 749

Appendix A Keystroke Guide for the TI-83 and TI-83 Plus/TI-84 Plus 758

Appendix B Proofs of Logarithmic Properties 767 Table of Symbols 768 Table of Measurement Abbreviations 768 Table of Properties 769 Table of Algebraic and Geometric Formulas 770

Solutions to You Try Its S1 Answers to Selected Exercises Index I1

A1


x

Preface The fourth edition of Algebra: Introductory and Intermediate examines the fundamental ideas of algebra. Recognizing that the basic principles of geometry are a necessary part of mathematics, we have also included a separate chapter on geometry (Chapter 3) and have integrated geometry topics, where appropriate, throughout the text. The text has been designed not only to meet the needs of the traditional college student, but also to serve the needs of returning students whose mathematical proficiency may have declined during years away from formal education. In this new edition of Algebra: Introductory and Intermediate, we have continued to integrate some of the approaches suggested by AMATYC. Each chapter opens with a photo and a reference to a mathematical application within the chapter. At the end of each section there are “Applying the Concepts” exercises, which include writing, synthesis, critical thinking, and challenge problems. At the end of each chapter there is a “Focus on Problem Solving,” which introduces students to various problem-solving strategies. This is followed by “Projects and Group Activities,” which can be used for cooperative-learning activities.

NEW! Changes to This Edition We have found that students who are taught division of a polynomial by a monomial as a separate topic are subsequently more successful in factoring a monomial from a polynomial. Therefore, we have added a new objective, “To divide a polynomial by a monomial,” to Section 4 of Chapter 6. In Section 11.3, the material on composition of functions has been expanded, and students are given more opportunities to apply the concept to applications. In the previous edition, complex numbers were presented in Section 9.3. In this edition, complex numbers have been moved to the last section of the chapter. This provides for a better flow of the material in Chapter 9 and places complex numbers immediately before Chapter 10, “Quadratic Equations,” where it is used extensively.


In Section 2 of Chapter 12, the introduction to logarithms has been rewritten. Motivating students to understand the need for logarithms is developed within the context of an application. This topic is presented at a slower pace to help students better understand and apply the concept of logarithm. The in-text examples are now highlighted by a prominent HOW TO bar. Students looking for a worked-out example can easily locate one of these problems. As another aid for students, more annotations have been added to the Examples provided in the paired Example/You Try It boxes. This will assist students in understanding what is happening in key steps of the solution to an exercise. Throughout the text, data problems have been updated to reflect current data and trends. Also, titles have been added to the application exercises in the exercise sets. These changes emphasize the relevance of mathematics and the variety of problems in real life that require mathematical analysis. The Chapter Summaries have been remodeled and expanded. Students are provided with definitions, rules, and procedures, along with examples of each. An objective reference and a page reference accompany each entry. We are confident that these will be valuable aids as students review material and study for exams. xi

Preface

In many chapters, the number of exercises in the Chapter Review Exercises has been increased. This will provide students with more practice on the concepts presented in the chapter. The calculator appendix has been expanded to include instruction on more functions of the graphing calculator. Notes entitled Integrating Technology appear throughout the book and many refer the student to this appendix. Annotated illustrations of both a scientific calculator and a graphing calculator appear on the inside back cover of this text. Feedback from users of the third edition informed us that the material on Cramer’s Rule and the material on conic sections were not covered in the majority of classes. Those who did present these topics presented only abbreviated coverage. Consequently, these topics are not included in the textbook in this fourth edition, which, of course, reduces the size of the text and lowers the cost to the students. However, material on Cramer’s Rule and conic sections is available online to instructors and students who use the text. Please see the Table of Contents, Sections 5.5 and 11.5.


xii

chapter

8

Chapter Opening Features

Rational Expressions

NEW! Chapter Opener New, motivating chapter opener photos and captions have been added, illustrating and referencing a specific application from the chapter. OBJECTIVES

Section 8.1

A B C

The

To simplify a rational expression To multiply rational expressions To divide rational expressions

students know of additional online resources at math.college.hmco.com/students.

Section 8.2

A B

To rewrite rational expressions in terms of a common denominator To add or subtract rational expressions

Objective-Specific Approach

Section 8.3

A

To simplify a complex fraction

Each chapter begins with a list of learning objectives that form the framework for a complete learning system. The objectives are woven throughout the text (i.e., Exercises, Prep Tests, Chapter Review Exercises, Chapter Tests, Cumulative Review Exercises) as well as through the print and multimedia ancillaries. This results in a seamless learning system delivered in one consistent voice.


Section 8.4 In order to monitor species that are or are becoming endangered, scientists need to determine the present population of that species. Scientists catch and tag a certain number of the animals and then release them. Later, a group of the animals from that same habitat is caught and the number tagged is counted. A proportion is used to estimate the total population size in that region, as shown in Exercise 27 on page 482. Tracking the tagged animals also assists scientists in learning more about the habits of that species.

A

To solve an equation containing fractions

Section 8.5

A B C

To solve a proportion To solve application problems To solve problems involving similar triangles

Section 8.6

A

To solve a literal equation for one of the variables

Section 8.7

A B

To solve work problems To use rational expressions to solve uniform motion problems

Section 8.8

A

at the bottom of the page lets

To solve variation problems

Need help? For online student resources, such as section quizzes, visit this textbook’s website at math.college.hmco.com/students.

Page 608 Page 451 PREP TEST Do these exercises to prepare for Chapter 11. b 2a

1.

Evaluate

for b 4 and a 2 .

2.

Given y x2 2x 1, find the value of y when x 2.

3.

Given fx x2 3x 2, find f 4.

4.

Evaluate pr r2 5 when r 2 h.

5.

Solve: 0 3x2 7x 6

6.

Solve by using the quadratic formula: 0 x2 4x 1

7.

Solve x 2y 4 for y.

8.

Find the domain and range of the relation 2, 4, 3, 5, 4, 6, 6, 5. Is the relation a function?

9.

Graph: x 2

Prep Test and Go Figure

The Go Figure problem that follows the Prep Test is a playful puzzle problem designed to engage students in problem solving.

y 4 2 –4

–2

0

2

4

x

–2 –4

GO FIGURE Each time the two hands of a certain standard, analog, 12-hour clock form a 180° angle, a bell chimes once. From noon today until noon tomorrow, how many chimes will be heard?



Prep Tests occur at the beginning of each chapter and test students on previously covered concepts that are required in the coming chapter. Answers are provided in the Answer Section. Objective references are also provided if a student needs to review specific concepts.

xiii

Objective B

To solve application problems

VIDEO & DVD

CD TUTOR

Aufmann Interactive Method (AIM)

539

Section 9.3 / Solving Equations Containing Radical Expressions

SSM

WEB

Hy

A right triangle contains one 90º angle. The side opposite the 90º angle is called the hypotenuse. The other two sides are called legs.

Leg

pot

enu

se

Leg

An Interactive Approach Pythagoras, a Greek mathematician, discovered that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs. This is called the Pythagorean Theorem.

Pythagoras

a

c

Algebra: Introductory and Intermediate uses an interactive style that provides a student with an opportunity to try a skill as it is presented. Each section is divided into objectives, and every objective contains one or more sets of matched-pair examples. The first example in each set is worked out; the second example, called “You Try It,” is for the student to work. By solving this problem, the student actively practices concepts as they are presented in the text.

b 2 2 2 c =a +b

(c. 580 B.C.–529 B.C.)

You Try It 3

A ladder 20 ft long is leaning against a building. How high on the building will the ladder reach when the bottom of the ladder is 8 ft from the building? Round to the nearest tenth.

Find the diagonal of a rectangle that is 6 cm long and 3 cm wide. Round to the nearest tenth.

20 f

t

Example 3

8 ft

Strategy

Your strategy


To find the distance, use the Pythagorean Theorem. The hypotenuse is the length of the ladder. One leg is the distance from the bottom of the ladder to the base of the building. The distance along the building from the ground to the top of the ladder is the unknown leg. Solution

c2 a2 b2 202 82 b2 400 64 b2 336 b2 3361/2 b21/2 336 b 18.3 b

You Try It 3 Strategy Your solution

• Pythagorean Theorem • Replace c by 20 and a by 8. • Solve for b.

To find the diagonal, use the Pythagorean Theorem. One leg is the length of the rectangle. The second leg is the width of the rectangle. The hypotenuse is the diagonal of the rectangle.

There are complete worked-out solutions to these examples in an appendix. By comparing their solution to the solution in the appendix, students obtain immediate feedback on, and reinforcement of, the concept.

Solution

1 2

• Raise each side to the power.

c2 a2 b2 c2 62 32 c2 36 9 c2 45

• a1/2 a

The distance is approximately 18.3 ft.

c212 4512 c 45 c 6.7

• Pythagorean Theorem • Replace a by 6 and b by 3. • Solve for c. 1 2

• Raise each side to the power. Solution on p. S29

• a1/2 a

The diagonal is approximately 6.7 cm.

Page 539

Page S29

Page xxiii

AIM for Success Student Preface

xiv

AIM for Success Welcome to Algebra: Introductory and Intermediate. As you begin this course, we know two important facts: (1) We want you to succeed. (2) You want to succeed. To do that requires an effort from each of us. For the next few pages, we are going to show you what is required of you to achieve that success and how you can use the features of this text to be successful.

Motivation

One of the most important keys to success is motivation. We can try to motivate you by offering interesting or important ways mathematics can benefit you. But, in the end, the motivation must come from you. On the first day of class, it is easy to be motivated. Eight weeks into the term, it is harder to keep that motivation.


This student ‘how to use this book’ preface explains what is required of a student to be successful and how this text has been designed to foster student success, including the Aufmann Interactive Method (AIM). AIM for Success can be used as a lesson on the first day of class or as a project for students to complete to strengthen their study skills. There are suggestions for teaching this lesson in the Instructor’s Resource Manual.

Problem Solving

283

Focus on Problem Solving

Focus on Problem Solving Find a Pattern


One approach to problem solving is to try to find a pattern. Karl Friedrich Gauss supposedly used this method to solve a problem that was given to his math class when he was in elementary school. As the story goes, his teacher wanted to grade some papers while the class worked on a math problem. The problem given to the class was to find the sum

Point of Interest

At the end of each chapter is a Focus on Problem Solving feature which introduces the student to various successful problem-solving strategies. Strategies such as drawing a diagram, applying solutions to other problems, working backwards, inductive reasoning, and trial and error are some of the techniques that are demonstrated.

1 2 3 4 100 Gauss quickly solved the problem by seeing a pattern. Here is what he saw. 101 101 101 101

Note that 1 100 101 2 99 101 3 98 101 4 97 101

1 2 3 4 97 98 99 100

Gauss noted that there were 50 sums of 101. Therefore, the sum of the first 100 natural numbers is 1 2 3 4 97 98 99 100 50101 5050

Karl Friedrich Gauss Karl Friedrich Gauss (1777–1855) has been called the “Prince of Mathematicians” by some historians. He applied his genius to many areas of mathematics and science. A unit of magnetism, the gauss, is named in his honor. Some types of electronic equipment (televisions, for instance) contain a degausser that controls magnetic fields.

Try to solve Exercises 1 to 6 by finding a pattern. 1. Find the sum 2 4 6 96 98 100. 2. Find the sum 1 3 5 97 99 101. 3. Find another method of finding the sum 1 3 5 97 99 101 given in the preceding exercise. 4. Find the sum Hint:

1 12

1 12

1 1 2 12

,

1 23

1 23

1 34 2 3

,

1 12

1 . 49 50

1 23

1 34

3 4


5. A polynomial number is a number that can be represented by arranging that number of dots in rows to form a geometric figure such as a triangle, square, pentagon, or hexagon. For instance, the first four triangular numbers, 3, 6, 10, and 15, are shown below. What are the next two triangular numbers?

3

2

2 3

1

1

2 points, 2 regions

4

3 points, 4 regions

2 6 7 5 8

3

1 4 4 points, 8 regions

5 points, ? regions

6

10

15

6. The following problem shows that checking a few cases does not always result in a conjecture that is true for all cases. Select any two points on a circle (see the drawing in the left margin) and draw a chord, a line connecting the points. The chord divides the circle into two regions. Now select three different points and draw chords connecting each of the three points with every other point. The chords divide the circle into four regions. Now select four points and connect each of the points with every other point. Make a conjecture as to the relationship between the number of regions and the number of points on the circle. Does your conjecture work for five points? six points?

Page 114 Page 283 114

Chapter 2 / First-Degree Equations and Inequalities

Example 1

You Try It 1

How many ounces of a silver alloy that costs $4 an ounce must be mixed with 10 oz of an alloy that costs $6 an ounce to make a mixture that costs $4.32 an ounce?

A gardener has 20 lb of a lawn fertilizer that costs $.80 per pound. How many pounds of a fertilizer that costs $.55 per pound should be mixed with this 20 lb of lawn fertilizer to produce a mixture that costs $.75 per pound?

Strategy

Your strategy

x oz $4/oz

The text features a carefully developed approach to problem solving that emphasizes the importance of strategy when solving problems. Students are encouraged to develop their own strategies—to draw diagrams, to write out the solution steps in words—as part of their solution to a problem. In each case, model strategies are presented as guides for students to follow as they attempt the “You Try It” problem. Having students provide strategies is a natural way to incorporate writing into the math curriculum.

10 oz $ 6/o z

• Ounces of $4 alloy: x

$4 alloy $6 alloy $4.32 mixture

Amount

Cost

Value

x 10 10 x

4 6 4.32

4x 610 4.3210 x

• The sum of the values before mixing equals the value after mixing.

Your solution

Solution

4x 610 4.3210 x 4x 60 43.2 4.32x 0.32x 60 43.2 0.32x 16.8 x 52.5 52.5 oz of the $4 silver alloy must be used.

Solution on p. S6



Problem-Solving Strategies

xv

224

Real Data and Applications

Chapter 4 / Linear Functions and Inequalities in Two Variables

A researcher may investigate the relationship between two variables by means of regression analysis, which is a branch of statistics. The study of the relationship between the two variables may begin with a scatter diagram, which is a graph of the ordered pairs of the known data.

Integrating

Technology See the appendix Keystroke Guide: Scatter Diagrams for instructions on using a graphing calculator to create a scatter diagram.

The following table shows randomly selected data for a recent Boston Marathon. Ages of participants 40 years old and older and their times (in minutes) are given. Age (x)

55

46

53

40

40

44

54

44

41

50

Time (y) 254

204

243

194

281

197

238

300

232

216

Time (in minutes)

The jagged portion of the horizontal axis in the figure at the right indicates that the numbers between 0 and 40 are missing.

One way to motivate an interest in mathematics is through applications. Wherever appropriate, the last objective of a section presents applications that require the student to use problem-solving strategies, along with the skills covered in that section, to solve practical problems. This carefully integrated applied approach generates student awareness of the value of algebra as a real-life tool.

y

The scatter diagram for these data is shown at the right. Each ordered pair represents the age and time of a participant. For instance, the ordered pair (53, 243) indicates that a 53-year-old participant ran the marathon in 243 min.

TA K E N O T E

Applications

300 200 100 0

40

45

50

55

x

Age

Example 6

You Try It 6

The grams of sugar and the grams of fiber in a 1-ounce serving of six breakfast cereals are shown in the table below. Draw a scatter diagram of these data.

According to the National Interagency Fire Center, the number of deaths in U.S. wildland fires is as shown in the table below. Draw a scatter diagram of these data.

Sugar (x)

Fiber (y)

Year

Number of Deaths

Wheaties

4

3

1998

14

Rice Krispies

3

0

1999

28

Total

5

3

2000

17

Life

6

2

2001

18

Kix

3

1

2002

23

Grape-Nuts

7

5

2003

29

Your strategy

To draw a scatter diagram: • Draw a coordinate grid with the horizontal axis representing the grams of sugar and the vertical axis the grams of fiber. • Graph the ordered pairs (4, 3), (3, 0), (5, 3), (6, 2), (3, 1), and (7, 5). Solution

Your solution

y

6 4 2 0

2

4

6

8

x

Number of deaths

Grams of fiber

8

y 30 25 20 15 10 5 0

'98 '99 '00 '01 '02 '03

Grams of sugar

x

Year

Solution on p. S12


Strategy

Applications are taken from many disciplines including astronomy, business, carpentry, chemistry, construction, Earth science, education, manufacturing, nutrition, real estate, and telecommunications.

Page 356 Page 224 356

117.

Chapter 6 / Polynomials

Astronomy The distance from Earth to Saturn is 8.86 108 mi. A satellite leaves Earth traveling at a constant rate of 1 105 mph. How long does it take for the satellite to reach Saturn? 8.86 103 h

118.

The Federal Government In 2004, the gross national debt was approximately 7 1012 dollars. How much would each American have to pay in order to pay off the debt? Use 3 108 as the number of citizens. 2.3 104 dollars

119.

Real Data Real data examples and exercises, identified by

Physics The mass of an electron is 9.109 1031 kg. The mass of a proton is 1.673 1027 kg. How many times heavier is a proton than an electron? 1.83664508 103 times heavier

120.

Geology The mass of Earth is 5.9 1024 kg. The mass of the sun is 2 1030 kg. How many times heavier is the sun than Earth? 3.3898305 105 times heavier

, ask students to analyze and solve

121.

Physics How many meters does light travel in 8 h? The speed of light is 3 108 ms. 8.64 1012 m

122.

Physics How many meters does light travel in 1 day? The speed of light is 3 108 ms. 2.592 1013 m

123.

Astronomy It took 11 min for the commands from a computer on Earth to travel to the rover Sojourner on Mars, a distance of 119 million miles. How fast did the signals from Earth to Mars travel? 1.081 107 mimin

124.

Measurement The weight of 31 million orchid seeds is 1 oz. Find the weight of one orchid seed.

Sojourner

3.2258065 108 oz

125. Physics A high-speed centrifuge makes 4 108 revolutions each minute. Find the time in seconds for the centrifuge to make one revolution.

APPLYING THE CONCEPTS Centrifuge

126.

Correct the error in each of the following expressions. Explain which rule or property was used incorrectly. a. x0 0 b. (x4)5 x9 c. x2 x3 x6

127. Simplify.

a. 1 1 (1 21)1 1 b. 2 2 (2 21)1 1

xvi

a.

8 5

b.

5 4


1.5 107 s


problems taken from actual situations. Students are often required to work with tables, graphs, and charts drawn from a variety of disciplines.

Student Pedagogy

543

Section 9.4 / Complex Numbers

9.4

Icons

Objective A

The

Complex Numbers To simplify a complex number

VIDEO & DVD

CD TUTOR

WEB

SSM

The radical expression 4 is not a real number because there is no real number whose square is 4. However, the solution of an algebraic equation is sometimes the square root of a negative number.

SSM

icons at each objective head remind students of the many and varied additional resources available for each objective.

For example, the equation x2 1 0 does not have a real number solution, because there is no real number whose square is a negative number.

x2 1 0 x2 1

Around the 17th century, a new number, called an imaginary number, was defined so that a negative number would have a square root. The letter i was chosen to represent the number whose square is 1.

Key Terms and Concepts

i2 1

Key terms, in bold, emphasize important terms. The key terms are also provided in a Glossary at the back of the text.

An imaginary number is defined in terms of i. Point of Interest The first written occurrence of an imaginary number was in a book published in 1545 by Hieronimo Cardan, where he wrote (in our modern notation) 5 15 . He went on to say that the number “is as refined as it is useless.” It was not until the 20th century that applications of complex numbers were found.

Key concepts are presented in orange boxes in order to highlight these important concepts and to provide for easy reference.

Point of Interest

Definition of a

If a is a positive real number, then the principal square root of negative a is the imaginary number i a . a i a

Here are some examples. 16 i16 4i 12 i12 2i3 21 i21 1 i1 i It is customary to write i in front of a radical to avoid confusing a i with ai .


These margin notes contain interesting sidelights about mathematics, its history, or its application.

Take Note These margin notes alert students to a point requiring special attention or are used to amplify the concept under discussion.

The real numbers and imaginary numbers make up the complex numbers.

Complex Number

A complex number is a number of the form a bi, where a and b are real numbers and i 1 . The number a is the real part of a bi , and the number b is the imaginary part.

TA K E N O T E The imaginary part of a complex number is a real number. As another example, the imaginary part of 6 8i is 8.

Examples of complex numbers are shown at the right.

Real Part

Imaginary Part a bi 3 2i 8 10i

Page 544 Page 543 544

Chapter 9 / Exponents and Radicals

Real Numbers a 0i

A real number is a complex number in which b 0.

Imaginary Numbers 0 bi

An imaginary number is a complex number in which a 0.

Complex numbers a bi

Study

HOW TO

Tip

Be sure you understand how to simplify expressions such as those in Example 1 and You Try It 1, as it is a prerequisite for solving quadratic equations in Chapter 10.

Simplify: 20 50

20 50 20 i50

Study Tips

• Write the complex number in the form a bi .

Example 1

You Try It 1

Simplify: 80

Simplify: 45

Solution

Your solution

These margin notes remind students of study skills presented in the AIM for Success; some notes provide page references to the original descriptions. They also provide students with reminders of how to practice good study habits.

Example 2

You Try It 2

HOW TO Examples

Simplify: 25 40

Simplify: 98 60

Solution

Your solution

4 5 i25 2

• Use the Product Property of Radicals to simplify each radical.

25 5i2

HOW TO examples use annotations to explain what is happening in key steps of the complete, worked-out solutions.

25 40 25 i40 25 i4 10 5 2i10 Solutions on p. S29

Objective B

Integrating

Technology See the appendix Keystroke Guide: Complex Numbers for instructions on using a graphing calculator to perform operations on complex numbers.

To add or subtract complex numbers

VIDEO & DVD

CD TUTOR

WEB

SSM

Addition and Subtraction of Complex Numbers

To add two complex numbers, add the real parts and add the imaginary parts. To subtract two complex numbers, subtract the real parts and subtract the imaginary parts. a bi c di a c b d i a bi c di a c b d i

HOW TO

Simplify: 3 7i 4 2i

3 7i 4 2i 3 4 7 2 i

• Subtract the real parts and subtract the imaginary parts of the complex numbers.

1 5i

Integrating Technology Copyright © Houghton Mifflin Company. All rights reserved.


80 i80 i16 5 4i5

These margin notes provide suggestions for using a calculator or refer the student to an appendix for more complete instructions on using a calculator.

xvii

682 14.

Exercises and Projects

Chapter 12 / Exponential and Logarithmic Functions

fx log32 x

15.

fx log2x 1

y

–2

fx log21 x y

4

2 –4

16.

y

4

4

2

0

2

4

x

–4

–2

2

0

2

x

4

–4

–2

0

–2

–2

–2

–4

–4

–4

2

x

4

Exercises The exercise sets of Algebra: Introductory and Intermediate emphasize skill building, skill maintenance, and applications. Concept-based writing or developmental exercises have been integrated within the exercise sets. Icons identify

APPLYING THE CONCEPTS Use a graphing calculator to graph the functions in Exercises 17 to 22. fx x log21 x

18.

1 fx log 2 x 1 2

y 4

20.

–2

2

4

x

–4

–2

23.

2

0

2

4

x

–4

–2

–4

–4

–4

21.

fx x2 10 lnx 1 4

2

2 2

4

x

–4

–2

0

2

4

appropriate writing

x

and calculator

, data analysis

,

exercises.

x 3 log 2 x 3 3

fx

y

4

0

22.

y 2 –4 2

4

–2

0

x

2

4

–2

–4

–4

–4

–6

M 4 0

5

10 15 20 25

2

4

s

−4 −8

The point with coordinates 25.1, 2 is on the graph. Write a sentence that describes the meaning of this ordered pair. s 55 50 45 40

The point with coordinates 4, 49 is on the graph. Write a sentence that describes the meaning of this ordered pair.

Included in each exercise set are Applying the Concepts that present extensions of topics, require analysis, or offer challenge problems. The writing exercises ask students to explain answers, write about a topic in the section, or research and report on a related topic.

x

–2

–2

24. Typing Without practice, the proficiency of a typist decreases. The equation s 60 7 lnt 1, where s is the typing speed in words per minute and t is the number of months without typing, approximates this decrease. a. Graph the equation. b.

0

–2

Astronomy Astronomers use the distance modulus of a star as a method of determining the star’s distance from Earth. The formula is M 5 logs 5, where M is the distance modulus and s is the star’s distance from Earth in parsecs. (One parsec 1.9 1013 mi) a. Graph the equation. b.

–2

–2

y

–2

y 4

2

0

fx x log 32 x

–4

x 2 log 2 x 1 2

fx

4

2 –4

19.

y

6 8 Months

10 12

t


17.

Page 388 Page 682 388

Chapter 6 / Polynomials

Projects and Group Activities Astronomical Distances and Scientific Notation

Astronomers have units of measurement that are useful for measuring vast distances in space. Two of these units are the astronomical unit and the light-year. An astronomical unit is the average distance between Earth and the sun. A light-year is the distance a ray of light travels in 1 year.

1. Light travels at a speed of 1.86 105 mis. Find the measure of 1 light-year in miles. Use a 365-day year. 2. The distance between Earth and the star Alpha Centauri is approximately 25 trillion miles. Find the distance between Earth and Alpha Centauri in light-years. Round to the nearest hundredth.

Projects and Group Activities The Projects and Group Activities featured at the end of each chapter can be used as extra credit or for cooperative learning activities. The projects cover various aspects of mathematics, including the use of calculators, collecting data from the Internet, data analysis, and extended applications.

3. The Coma cluster of galaxies is approximately 2.8 108 light-years from Earth. Find the distance, in miles, from the Coma cluster to Earth. Write the answer in scientific notation.

Gemini

5. One light-year is equal to approximately how many astronomical units? Round to the nearest thousand.

Point of Interest In November 2001, the Hubble Space Telescope took photos of the atmosphere of a planet orbiting a star 150 light-years from Earth in the constellation Pegasus. The planet is about the size of Jupiter and orbits close to the star HD209458. It was the first discovery of an atmosphere around a planet outside our solar system.

Planet

Distance from the Sun (in kilometers)

Mass (in kilograms)

Earth

1.50 × 108

5.97 × 1024

Jupiter

7.79 × 108

1.90 × 1027

Mars

2.28 × 108

6.42 × 1023

Mercury

5.79 × 107

3.30 × 1023

Neptune

4.50 × 109

1.02 × 1026

Pluto

5.87 × 109

1.25 × 1022

Saturn

9

1.43 × 10

5.68 × 1026

Uranus

2.87 × 109

8.68 × 1025

Venus

1.08 × 108

4.87 × 1024

6. Arrange the planets in order from closest to the sun to farthest from the sun. 7. Arrange the planets in order from the one with the greatest mass to the one with the least mass. Jupiter

xviii

8. Write a rule for ordering numbers written in scientific notation.


Shown below are data on the planets in our solar system. The planets are listed in alphabetical order.


4. One astronomical unit (A.U.) is 9.3 107 mi. The star Pollux in the constellation Gemini is 1.8228 1012 mi from Earth. Find the distance from Pollux to Earth in astronomical units.

Chapter 2 Summary

149

End of Chapter

Chapter 2 Summary Key Words

Examples

An equation expresses the equality of two mathematical expressions. [2.1A, p. 73]

3 24x 5 x 4 is an equation.

Chapter Summary At the end of each chapter there is a Chapter Summary that includes Key Words, Essential Rules and Procedures, and an example of each. Each entry includes an objective reference and a page reference indicating where the concept is introduced. These chapter summaries provide a single point of reference as the student prepares for a test.

Page 149

Essential Rules and Procedures Addition Property of Equations [2.1B, p. 74]

The same number can be added to each side of an equation without changing the solution of the equation. If a b, then a c b c.

Examples x 5 3 x 5 5 3 5 x 8

Page 150

Page 152 152


Chapter Review Exercises Chapter Review Exercises are found at the end of each chapter. These exercises are selected to help the student integrate all of the topics presented in the chapter.

Chapter 2 Review Exercises 1.

Solve: 3t 3 2t 7t 15

Solve: 3x 7 2 5 xx [2.5A] 3

2.

6 [2.2B]

3.

Is 3 a solution of 5x 2 4x 5?

Solve: x 4 5

4.

9 [2.1B]

No [2.1A]

Page 155 Chapter 2 Test

Chapter Test Each Chapter Test is designed to simulate a possible test of the material in the chapter.

155

Chapter 2 Test 1.

Solve: x 2 4

Solve: b

2.

2 [2.1B]

3.

1 8

3 5 Solve: y 4 8 5

3 5 4 8

[2.1B]

Solve: 3x 5 7

4.

4

[2.2A]

[2 1C]

Page 157


Cumulative Review Exercises

Cumulative Review Exercises Cumulative Review Exercises, which appear at the end of each chapter (beginning with Chapter 2), help students maintain skills learned in previous chapters.

Cumulative Review Exercises 1.

Subtract: 6 (20) 8

2.

Multiply: (2)(6)(4) 48 [1.1D]

6 [1.1C]

3.

Subtract:

5 7 6 16

19

The answers to all Chapter Review Exercises, all Chapter Test exercises, and all Cumulative Review Exercises are given in the Answer Section. Along with the answer, there is a reference to the objective that pertains to each exercise.

157

Simplify: 42

4.

54 [1 2E]

3 2

3

Page A6 CUMULATIVE REVIEW EXERCISES 1. 6 [1.1C]

2. 48 [1.1D]

3.

19 48

[1.2C]

8. 5a 4b [1.4B]

9. 2x [1.4C]

13. 6x 34 [1.4D]

14. A B {4, 0} [1.5A]

4. 54 [1.2E]

10. 36y [1.4C] 15.

11.

2x2

5.

49 40

[1.3A]

6x 4 [1.4D]

−5 −4 −3 −2 − 1 0 1 2 3 4 5

6. 6 [1.4A]

7. 17x [1.4B]

12. 4x 14 [1.4D] [1.5C]

16. Yes [2.1A]

1

xix

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Preface

xxi

Instructor Resources Algebra: Introductory and Intermediate has a complete set of support materials for the instructor. Instructor’s Annotated Edition This edition contains a replica of the student text and additional items just for the instructor. Online Instructor’s Resource Manual with Solutions The Instructor’s Resource Manual with Solutions contains worked-out solutions for all exercises in the text. It also contains suggested Course Sequences and a printout of the AIM for Success PowerPoint slide show. It is available on the ClassPrep CD and at the Online Teaching Center. Online Instructor’s Test Bank with Chapter Tests This resource contains a static version of the HM Testing files. It also contains eight ready-to-use Chapter Tests per chapter. All resources are also available on the ClassPrep CD and at the Online Teaching Center. HM ClassPrep™ with HM Testing (powered by Diploma™) HM ClassPrep offers a combination of two class management tools including supplements and text-specific resources for the instructor. HM Testing offers instructors a flexible and powerful tool for test generation and test management. Now supported by the Brownstone Research Group’s market-leading Diploma software, this new version of HM Testing significantly improves on functionality and ease of use by offering all the tools needed to create, author, deliver, and customize multiple types of tests—including authoring and editing algorithmic questions. Online Teaching Center For an abundance of instructor resources, visit the free Houghton Mifflin Teaching Center on our website, college.hmco.com/pic/ aufmannAIAI4e.


Blackboard®, WebCT®, and eCollege® Houghton Mifflin can provide you with valuable content to include in your existing Blackboard, WebCT, and eCollege systems. This text-specific content enables instructors to teach all or part of their course online. Contact your Houghton Mifflin sales rep for cartridge availability. TeamUP Integration Services TeamUP, our integration program, offers flexible, personalized training and consultative services by phone, online, or on campus. The TeamUP Integration Team will: • Show you how to use our products. • Provide ideas and best practices for incorporating all elements of our text and technology program into your course. • Customize our programs or products to achieve your teaching objectives. • Provide technical assistance. • Link you to faculty development opportunities. Visit teamup.college.hmco.com for more information, or contact your Houghton Mifflin sales representative to schedule one of our customized programs.

NEW! Eduspace® Eduspace, powered by Blackboard, is Houghton Mifflin’s online learning tool. Eduspace makes it easy for instructors to create all or part of a course online. Homework exercises, quizzes, tests, tutorials, and supplemental study materials all come ready-to-use. Instructors can choose to use the content as is or modify it, and they can even add their own. Visit www.eduspace.com for more information.

NEW! HM Assess for Mathematics HM Assess is a new diagnostic assessment tool from Houghton Mifflin that tests core concepts in specific courses and provides students with individualized study paths for self-remediation. These paths are carefully designed to offer self-study options and to appeal to a variety of learning styles with concept reviews, interactive lessons, similar examples, and additional practice exercises. Instructors can use HM Assess to quickly gauge which

Preface

students are in jeopardy and which concepts require additional review. Both Course Assessments and Chapter Assessments are available. HM Assess is offered as part of Eduspace. Visit hmassess.college.hmco.com for more information.

Student Resources Student Solutions Manual The Student Solutions Manual contains complete solutions to odd-numbered exercises in the text. Math Study Skills Workbook by Paul D. Nolting This workbook is designed to reinforce skills and minimize frustration for students in any math class, lab, or study skills course. It offers a wealth of study tips and sound advice on note taking, time management, and reducing math anxiety. In addition, numerous opportunities for self-assessment enable students to track their own progress. Eduspace® Eduspace, powered by Blackboard, is Houghton Mifflin’s online learning tool. Eduspace is a text-specific, web-based learning environment offering students a combination of practice exercises, multimedia tutorials, video explanations, online algorithmic homework, and more. Specific content is available 24 hours a day to help you succeed in your course. HM mathSpace® Tutorial CD-ROM For students who prefer the portability of a CD-ROM, this tutorial provides opportunities for self-paced review and practice with algorithmically generated exercises and step-by-step solutions. SMARTHINKING® Houghton Mifflin’s unique partnership with SMARTHINKING brings students real-time, online tutorial support when they need it most. Using state-of-the-art whiteboard technology and feedback tools, students interact and communicate with “e-structors.” These specially trained tutors guide students through the learning and problem solving process without providing answers or rewriting a student’s work. SMARTHINKING offers three levels of service.* • Live Tutorial Help provides real-time, one-on-one instruction. • Questions Any Time allows students to e-mail questions to a tutor outside of the scheduled tutorial sessions and receive a reply, usually within 24 hours. • Independent Study Resources connects students around-the-clock to additional educational resources, ranging from interactive websites to Frequently Asked Questions. Visit smarthinking.com for more information. *Limits apply; terms and hours of SMARTHINKING service are subject to change. Houghton Mifflin Instructional Videos and DVDs Text-specific videos and DVDs, hosted by Dana Mosely, cover all sections of the text and provide a valuable resource for further instruction and review. Online Study Center For an abundance of student resources, visit the free Houghton Mifflin Study Center on our website, college.hmco.com/pic/ aufmannAIAI4e.

Acknowledgments The authors would like to thank the people who have reviewed this manuscript and provided many valuable suggestions. Donna Foster, Piedmont Technical College William Graesser, Ivy Tech State College Anne Haney Tim R. McBride, Spartanburg Technical College Michael McComas, Marshall Community and Technical College Linda J. Murphy, Northern Essex Community College


xxii

AIM for Success Welcome to Algebra: Introductory and Intermediate. As you begin this course, we know two important facts: (1) We want you to succeed. (2) You want to succeed. To do that requires an effort from each of us. For the next few pages, we are going to show you what is required of you to achieve that success and how you can use the features of this text to be successful.

Motivation

TA K E N O T E Motivation alone will not lead to success. For instance, suppose a person who cannot swim is placed in a boat, taken out to the middle of a lake, and then thrown overboard. That person has a lot of motivation but there is a high likelihood the person will drown without some help. Motivation gives us the desire to learn but is not the same as learning.

Commitment

One of the most important keys to success is motivation. We can try to motivate you by offering interesting or important ways mathematics can benefit you. But, in the end, the motivation must come from you. On the first day of class, it is easy to be motivated. Eight weeks into the term, it is harder to keep that motivation. To stay motivated, there must be outcomes from this course that are worth your time, money, and energy. List some reasons you are taking this course.

Although we hope that one of the reasons you listed was an interest in mathematics, we know that many of you are taking this course because it is required to graduate, it is a prerequisite for a course you must take, or because it is required for your major. Although you may not agree that this course is necessary, it is! If you are motivated to graduate or complete the requirements for your major, then use that motivation to succeed in this course. Do not become distracted from your goal to complete your education! To be successful, you must make a commitment to succeed. This means devoting time to math so that you achieve a better understanding of the subject. List some activities (sports, hobbies, talents such as dance, art, or music) that you enjoy and at which you would like to become better.


ACTIVITY

TIME SPENT

TIME WISHED SPENT

Thinking about these activities, put the number of hours that you spend each week practicing these activities next to the activity. Next to that number, indicate the number of hours per week you would like to spend on these activities. Whether you listed surfing or sailing, aerobics or restoring cars, or any other activity you enjoy, note how many hours a week you spend doing it. To succeed in math, you must be willing to commit the same amount of time. Success requires some sacrifice.

The “I Can’t Do Math” Syndrome

There may be things you cannot do, such as lift a two-ton boulder. You can, however, do math. It is much easier than lifting the two-ton boulder. When you first xxiii

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learned the activities you listed above, you probably could not do them well. With practice, you got better. With practice, you will be better at math. Stay focused, motivated, and committed to success. It is difficult for us to emphasize how important it is to overcome the “I Can’t Do Math” Syndrome. If you listen to interviews of very successful athletes after a particularly bad performance, you will note that they focus on the positive aspect of what they did, not the negative. Sports psychologists encourage athletes to always be positive—to have a “Can Do” attitude. Develop this attitude toward math.

Strategies for Success

Textbook Review Here’s how:

Right now, do a 15-minute “textbook review” of this book.

First, read the table of contents. Do it in three minutes or less. Next, look through the entire book, page by page. Move quickly. Scan titles, look at pictures, notice diagrams. A textbook review shows you where a course is going. It gives you the big picture. That’s useful because brains work best when going from the general to the specific. Getting the big picture before you start makes details easier to recall and understand later on. Your textbook review will work even better if, as you scan, you look for ideas or topics that are interesting to you. List three facts, topics, or problems that you found interesting during your textbook review.

The idea behind this technique is simple: It’s easier to work at learning material if you know it’s going to be useful to you. Not all the topics in this book will be “interesting” to you. But that is true of any subject. Surfers find that on some days the waves are better than others, musicians find some music more appealing than other music, computer gamers find some computer games more interesting than others, car enthusiasts find some cars more exciting than others. Some car enthusiasts would rather have a completely restored 1957 Chevrolet than a new Ferrari. Know the Course Requirements To do your best in this course, you must know exactly what your instructor requires. Course requirements may be stated in a syllabus, which is a printed outline of the main topics of the course, or they may be presented orally. When they are listed in a syllabus or on other printed pages, keep them in a safe place. When they are presented orally, make sure to take complete notes. In either case, it is important that you understand them completely and follow them exactly. Be sure you know the answer to each of the following questions. 1. 2. 3. 4. 5. 6.

What is your instructor’s name? Where is your instructor’s office? At what times does your instructor hold office hours? Besides the textbook, what other materials does your instructor require? What is your instructor’s attendance policy? If you must be absent from a class meeting, what should you do before returning to class? What should you do when you return to class?


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7. What is the instructor’s policy regarding collection or grading of homework assignments? 8. What options are available if you are having difficulty with an assignment? Is there a math tutoring center? 9. If there is a math lab at your school, where is it located? What hours is it open? 10. What is the instructor’s policy if you miss a quiz? 11. What is the instructor’s policy if you miss an exam? 12. Where can you get help when studying for an exam? Remember: Your instructor wants to see you succeed. If you need help, ask! Do not fall behind. If you are running a race and fall behind by 100 yards, you may be able to catch up but it will require more effort than had you not fallen behind. TA K E N O T E Besides time management, there must be realistic ideas of how much time is available. There are very few people who can successfully work full-time and go to school full-time. If you work 40 hours a week, take 15 units, spend the recommended study time given at the right, and sleep 8 hours a day, you will use over 80% of the available hours in a week. That leaves less than 20% of the hours in a week for family, friends, eating, recreation, and other activities.


Monday 7–8 a.m. 8–9 a.m. 9–10 a.m. 10–11 a.m. 11–12 p.m. 12–1 p.m. 1–2 p.m. 2–3 p.m. 3–4 p.m. 4–5 p.m. 5–6 p.m. 6–7 p.m. 7–8 p.m. 8–9 p.m. 9–10 p.m. 10–11 p.m. 11–12 a.m.

Time Management We know that there are demands on your time. Family, work, friends, and entertainment all compete for your time. We do not want to see you receive poor job evaluations because you are studying math. However, it is also true that we do not want to see you receive poor math test scores because you devoted too much time to work. When several competing and important tasks require your time and energy, the only way to manage the stress of being successful at both is to manage your time efficiently. Instructors often advise students to spend twice the amount of time outside of class studying as they spend in the classroom. Time management is important if you are to accomplish this goal and succeed in school. The following activity is intended to help you structure your time more efficiently. List the name of each course you are taking this term, the number of class hours each course meets, and the number of hours you should spend studying each subject outside of class. Then fill in a weekly schedule like the one printed below. Begin by writing in the hours spent in your classes, the hours spent at work (if you have a job), and any other commitments that are not flexible with respect to the time that you do them. Then begin to write down commitments that are more flexible, including hours spent studying. Remember to reserve time for activities such as meals and exercise. You should also schedule free time. Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

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We know that many of you must work. If that is the case, realize that working 10 hours a week at a part-time job is equivalent to taking a three-unit class. If you must work, consider letting your education progress at a slower rate to allow you to be successful at both work and school. There is no rule that says you must finish school in a certain time frame. Schedule Study Time As we encouraged you to do by filling out the time management form on the previous page, schedule a certain time to study. You should think of this time the way you would the time for work or class—that is, reasons for missing study time should be as compelling as reasons for missing work or class. “I just didn’t feel like it” is not a good reason to miss your scheduled study time. Although this may seem like an obvious exercise, list a few reasons you might want to study.

Of course we have no way of knowing the reasons you listed, but from our experience one reason given quite frequently is “To pass the course.” There is nothing wrong with that reason. If that is the most important reason for you to study, then use it to stay focused. One method of keeping to a study schedule is to form a study group. Look for people who are committed to learning, who pay attention in class, and who are punctual. Ask them to join your group. Choose people with similar educational goals but different methods of learning. You can gain insight from seeing the material from a new perspective. Limit groups to four or five people; larger groups are unwieldy.

1. Test each other by asking questions. Each group member might bring two or three sample test questions to each meeting. 2. Practice teaching each other. Many of us who are teachers learned a lot about our subject when we had to explain it to someone else. 3. Compare class notes. You might ask other students about material in your notes that is difficult for you to understand. 4. Brainstorm test questions. 5. Set an agenda for each meeting. Set approximate time limits for each agenda item and determine a quitting time. And finally, probably the most important aspect of studying is that it should be done in relatively small chunks. If you can study only three hours a week for this course (probably not enough for most people), do it in blocks of one hour on three separate days, preferably after class. Three hours of studying on a Sunday is not as productive as three hours of paced study.

Text Features That Promote Success

There are 12 chapters in this text. Each chapter is divided into sections, and each section is subdivided into learning objectives. Each learning objective is labeled with a letter from A to G.


There are many ways to conduct a study group. Begin with the following suggestions and see what works best for your group.

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Preparing for a Chapter Before you begin a new chapter, you should take some time to review previously learned skills. There are two ways to do this. The first is to complete the Cumulative Review Exercises, which occurs after every chapter (except Chapter 1). For instance, turn to page 341. The questions in this review are taken from the previous chapters. The answers for all these exercises can be found on page A17. Turn to page A17 now and locate the answers for the Chapter 5 Cumulative Review Exercises. After the answer to the first exercise, which is 610, you will see the objective reference [1.2F]. This means that this question was taken from Chapter 1, Section 2, Objective F. If you missed this question, you should return to that objective and restudy the material. A second way of preparing for a new chapter is to complete the Prep Test. This test focuses on the particular skills that will be required for the new chapter. Turn to page 296 to see a Prep Test. The answers for the Prep Test are the first set of answers in the answer section for a chapter. Turn to page A14 to see the answers for the Chapter 5 Prep Test. Note that an objective reference is given for each question. If you answer a question incorrectly, restudy the objective from which the question was taken. Before the class meeting in which your professor begins a new section, you should read each objective statement for that section. Next, browse through the objective material, being sure to note each word in bold type. These words indicate important concepts that you must know in order to learn the material. Do not worry about trying to understand all the material. Your professor is there to assist you with that endeavor. The purpose of browsing through the material is so that your brain will be prepared to accept and organize the new information when it is presented to you.


Turn to page 3. Write down the title of the first objective in Section 1.1. Write down the words under the title of the objective that are in bold print. It is not necessary for you to understand the meaning of these words. You are in this class to learn their meaning.

Math Is Not a Spectator Sport To learn mathematics you must be an active participant. Listening and watching your professor do mathematics is not enough. Mathematics requires that you interact with the lesson you are studying. If you filled in the blanks above, you were being interactive. There are other ways this textbook has been designed to help you be an active learner. Annotated Examples The HOW TO feature indicates an example with explanatory remarks to the right of the work. Using paper and pencil, you should work along as you go through the example.

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HOW TO

3x 2 4 3x 6 3x 6 3 3 x 2 The solution set is x x 2.

Solve: 3x 2 4

3x 2 4 3x 6 3x 6 3 3 x 2

• Subtract 2 from each side of the inequality. • Divide each side of the inequality by the coefficient 3.

The solution set is x x 2. Page 127

When you complete the example, get a clean sheet of paper. Write down the problem and then try to complete the solution without referring to your notes or the book. When you can do that, move on to the next part of the objective. Leaf through the book now and write down the page numbers of two other occurrences of a HOW TO example. You Try Its One of the key instructional features of this text is the paired examples. Notice that in each example box, the example on the left is completely worked out and the “You Try It” example on the right is not. Study the workedout example carefully by working through each step. Then work the You Try It. If you get stuck, refer to the page number at the end of the example, which directs you to the place where the You Try It is solved —a complete worked-out solution is provided. Try to use the given solution to get a hint for the step you are stuck on. Then try to complete your solution. Example 2

You Try It 2

Solve: 3x 5 3 2 3x 1

Solve: 5x 2 4 3 x 2

Solution

Your solution

3x 5 3 2 3x 1

3x 5 3 6x 2 3x 5 1 6x 9x 5 1 9x 6 9x 6 9 9 2 x 3

x x

2 3

5x 2 4 3 x 2

5x 2 4 3x 6 5x 2 10 3x 8x 2 10 8x 12 8x 12 8 8 3 x 2

,

3 2

Solution on p. S7

TA K E N O T E There is a strong connection between reading and being a successful student in math or any other subject. If you have difficulty reading, consider taking a reading course. Reading is much like other skills. There are certain things you can learn that will make you a better reader.

When you have completed your solution, check your work against the solution we provided. (Turn to page S7 to see the solution of You Try It 2.) Be aware that frequently there is more than one way to solve a problem. Your answer, however, should be the same as the given answer. If you have any question as to whether your method will “always work,” check with your instructor or with someone in the math center. Browse through the textbook and write down the page numbers where two other paired example features occur. Remember: Be an active participant in your learning process. When you are sitting in class watching and listening to an explanation, you may think that you understand. However, until you actually try to do it, you will have no confirmation of the new knowledge or skill. Most of us have had the experience of sitting in class thinking we knew how to do something only to get home and realize that we didn’t.


Page 128

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Word Problems Word problems are difficult because we must read the problem, determine the quantity we must find, think of a method to do that, and then actually solve the problem. In short, we must formulate a strategy to solve the problem and then devise a solution. Note in the paired example below that part of every word problem is a strategy and part is a solution. The strategy is a written description of how we will solve the problem. In the corresponding You Try It, you are asked to formulate a strategy. Do not skip this step, and be sure to write it out.

Example 6

You Try It 6

Find three consecutive positive odd integers whose sum is between 27 and 51.

An average score of 80 to 89 in a history course receives a B. Luisa Montez has grades of 72, 94, 83, and 70 on four exams. Find the range of scores on the fifth exam that will give Luisa a B for the course.

Strategy

Your strategy

To find the three integers, write and solve a compound inequality using n to represent the first odd integer.

To find the scores, write and solve an inequality. Let N be the score on the last test.

Solution

Your solution 72 94 83 70 N 89 80 5 319 N 80 89 5 319 N 5 80 5 5 89

5 400 319 N 445 400 319 319 N 319 445 319 81 N 126

Lower limit upper limit of the sum sum of the sum 27 n n 2 n 4 51 27 3n 6 51 27 6 3n 6 6 51 6 21 3n 45 21 3n 45 3 3 3 7 n 15 The three odd integers are 9, 11, and 13; or 11, 13, and 15; or 13, 15, and 17.

The range of scores to get a B is 81 N 100. Solutions on p. S8

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TA K E N O T E


If a rule has more than one part, be sure to make a notation to that effect.

I can add the same number to both sides of an inequality and not change the solution set.

Rule Boxes Pay special attention to rules placed in boxes. These rules give you the reasons certain types of problems are solved the way they are. When you see a rule, try to rewrite the rule in your own words. When solving an inequality, we use the Addition and Multiplication Properties of Inequalities to rewrite the inequality in the form variable constant or in the form variable constant. The Addition Property of Inequalities

If a b , then a c b c. If a b , then a c b c.

TA K E N O T E If you are working at home and need assistance, there is online help available at math.college.hmco.com/ students, at this text’s website.

Page 125

Chapter Exercises When you have completed studying an objective, do the exercises in the exercise set that correspond with that objective. The exercises are labeled with the same letter as the objective. Math is a subject that needs to be learned in small sections and practiced continually in order to be mastered. Doing all of the exercises in each exercise set will help you master the problemsolving techniques necessary for success. As you work through the exercises for an objective, check your answers to the odd-numbered exercises with those in the back of the book.

AIM for Success

Preparing for a Test There are important features of this text that can be used to prepare for a test. • Chapter Summary • Chapter Review Exercises • Chapter Test After completing a chapter, read the Chapter Summary. (See page 285 for the Chapter 4 Summary.) This summary highlights the important topics covered in the chapter. The page number following each topic refers you to the page in the text on which you can find more information about the concept. Following the Chapter Summary are Chapter Review Exercises (see page 288) and a Chapter Test (see page 291). Doing the review exercises is an important way of testing your understanding of the chapter. The answer to each review exercise is given at the back of the book, along with its objective reference. After checking your answers, restudy any objective from which a question you missed was taken. It may be helpful to retry some of the exercises for that objective to reinforce your problem-solving techniques. The Chapter Test should be used to prepare for an exam. We suggest that you try the Chapter Test a few days before your actual exam. Take the test in a quiet place and try to complete the test in the same amount of time you will be allowed for your exam. When taking the Chapter Test, practice the strategies of successful test takers: (1) scan the entire test to get a feel for the questions; (2) read the directions carefully; (3) work the problems that are easiest for you first; and perhaps most importantly, (4) try to stay calm. When you have completed the Chapter Test, check your answers. If you missed a question, review the material in that objective and rework some of the exercises from that objective. This will strengthen your ability to perform the skills in that objective. Is it difficult to be successful? YES! Successful music groups, artists, professional athletes, chefs, and Write your major here have to work very hard to achieve their goals. They focus on their goals and ignore distractions. The things we ask you to do to achieve success take time and commitment. We are confident that if you follow our suggestions, you will succeed.


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1

Real Numbers and Variable Expressions

OBJECTIVES

Section 1.1

A B C D E

To use inequality symbols with integers To find the additive inverse and absolute value of a number To add or subtract integers To multiply or divide integers To solve application problems

Section 1.2


A B

When you take a multiple-choice test such as a class exam, the ACT, or the SAT, there is usually a point system for scoring your answers. Correct answers receive a positive number of points, and incorrect answers receive a negative number of points. For the ACT and the SAT, you will score higher if you leave a question blank when you are unsure of the answer. An unanswered question will cause fewer points to be deducted from your score; sometimes it will not cost you any points at all. Exercises 182 and 183 on page 15 show how professors can adjust the grading systems on multiple-choice exams to discourage students from guessing randomly.

C D E F G

To write a rational number as a decimal To convert among percents, fractions, and decimals To add or subtract rational numbers To multiply or divide rational numbers To evaluate exponential expressions To simplify numerical radical expressions To solve application problems

Section 1.3

A

To use the Order of Operations Agreement to simplify expressions

Section 1.4

A B C D E

To evaluate a variable expression To simplify a variable expression using the Properties of Addition To simplify a variable expression using the Properties of Multiplication To simplify a variable expression using the Distributive Property To translate a verbal expression into a variable expression

Section 1.5 Need help? For online student resources, such as section quizzes, visit this textbook’s website at math.college.hmco.com/students.

A B C

To write a set using the roster method To write a set using set-builder notation To graph an inequality on the number line

PREP TEST

1.

What is 127.1649 rounded to the nearest hundredth?

2.

Add: 49.147 5.96

3.

Subtract: 5004 487

4.

Multiply: 407 28

5.

Divide: 456 19

6.

What is the smallest number that both 8 and 12 divide evenly into?

7.

What is the greatest number that divides evenly into both 16 and 20?

8.

Without using 1, write 21 as a product of two whole numbers.

9.

Represent the shaded portion of the figure as a fraction.

GO FIGURE If you multiply the first 20 natural numbers (1 2 3 4 5 . . . 17 18 19 20), how many zeros will be at the end of the number?


Do these exercises to prepare for Chapter 1.

Section 1.1 / Integers

1.1 Objective A

3

Integers To use inequality symbols with integers

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It seems to be a human characteristic to group similar items. For instance, nutritionists classify foods according to food groups: pasta, crackers, and rice are among the foods in the bread group. Mathematicians likewise place objects with similar properties in sets and use braces to surround a list of the objects in the set, which are called elements. The numbers that we use to count elements, such as the number of people at a baseball game or the number of horses on a ranch, have similar characteristics. These numbers are the natural numbers. Natural numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, . . .} The natural numbers alone do not provide all the numbers that are useful in applications. For instance, a meteorologist needs numbers below zero and above zero. Integers {. . . , 5,4,3,2,1, 0, 1, 2, 3, 4, 5, . . .} − 5 − 4 − 3 −2 − 1 0 1 2 3 4 5

Negative Positive integers Zero integers

Point of Interest


The Alexandrian astronomer Ptolemy began using omicron, 0, the first letter of the Greek word that means “nothing,” as the symbol for zero in A.D. 150. It was not until the 13th century, however, that Fibonacci introduced 0 to the Western world as a placeholder so that we could distinguish, for example, 45 from 405.

Each integer can be shown on a number line. The graph of an integer is shown by placing a heavy dot on the number line directly above the number. The graphs of 3 and 4 are shown on the number line at the left. The integers to the left of zero are negative integers. The integers to the right of zero are positive integers. Zero is neither a positive nor a negative integer. Consider the sentences below. The quarterback threw the football and the receiver caught it. An accountant purchased a calculator and placed it in a briefcase. In the first sentence, it means football; in the second sentence, it means calculator. In language, the word it can stand for many different objects. Similarly, in mathematics, a letter of the alphabet can be used to stand for a number. Such a letter is called a variable. Variables are used in the next definition. Definition of Inequality Symbols

If a and b are two numbers and a is to the left of b on the number line, then a is less than b. This is written a b. If a and b are two numbers and a is to the right of b on the number line, then a is greater than b. This is written a b.

There are also inequality symbols for less than or equal to ( ) and greater than or equal to (). For instance, 6 6 because 6 6.

7 15 because 7 15.

It is convenient to use a variable to represent, or stand for, any one of the elements of a set. For instance, the statement “x is an element of the set {0, 2, 4, 6}” means that x can be replaced by 0, 2, 4, or 6. The symbol for “is an element of” is ; the symbol for “is not an element of” is . For example, 2 {0, 2, 4, 6}

6 {0, 2, 4, 6}

7 {0, 2, 4, 6}

4

Chapter 1 / Real Numbers and Variable Expressions

Example 1

You Try It 1

Let x {6, 2, 0}. For which values of x is the inequality x 2 a true statement?

Let y {5, 1, 5}. For which values of y is the inequality y 1 a true statement?

Solution

Your solution

Replace x by each element of the set and determine whether the inequality is true. x 2 6 2 True. 6 2 2 2 True. 2 2 0 2 False. The inequality is true for 6 and 2. Solution on p. S1

5

5

−5 − 4 − 3 − 2 − 1 0 1 2 3 4 5

TA K E N O T E The distance from 0 to 5 is 5; 5 5. The distance from 0 to 5 is 5; 5 5.

Point of Interest

To find the additive inverse and absolute value of a number

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On the number line, the numbers 5 and 5 are the same distance from zero but on opposite sides of zero. The numbers 5 and 5 are called opposites or additive inverses of each other. (See the number line at the left.) The opposite (or additive inverse) of 5 is 5. The opposite of 5 is 5. The symbol for opposite is . (5) means the opposite of positive 5. (5) means the opposite of negative 5.

(5) 5 (5) 5

The absolute value of a number is its distance from zero on the number line. The symbol for absolute value is two vertical bars, .

The definition of absolute value that we have given in the box is written in what is called “rhetorical style.” That is, it is written without the use of variables. This is how all mathematics was written prior to the Renaissance. During that period, from the 14th to the 16th century, the idea of expressing a variable symbolically was developed.

Absolute Value

The absolute value of a positive number is the number itself. The absolute value of zero is zero. The absolute value of a negative number is the opposite of the negative number.

HOW TO

9 9 0 0 7 7

Evaluate: 12 12 12

• The absolute value sign does not affect the negative sign in front of the absolute value sign.

Example 2

You Try It 2

Let a {12, 0, 4}. Find the additive inverse of a and the absolute value of a for each element of the set.

Let z {11, 0, 8}. Find the additive inverse of z and the absolute value of z for each element of the set.

Solution

Your solution

Replace a by each element of the set. a (12) 12 (0) 0 (4) 4

a 12 12 0 0 4 4 Solution on p. S1


Objective B

5


Objective C

To add or subtract integers

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A number can be represented anywhere along the number line by an arrow. A positive number is represented by an arrow pointing to the right, and a negative number is represented by an arrow pointing to the left. The size of the number is represented by the length of the arrow. −4

+5 – 10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1

0

1

2

3

4

5

6

7

8

9 10

Addition of integers can be shown on the number line. To add integers, start at zero and draw an arrow representing the first number. At the tip of the first arrow, draw a second arrow representing the second number. The sum is below the tip of the second arrow. TA K E N O T E Each number in a sum is called an addend. For instance, 4 and 2 are addends in the sum 4 2 6.

426

+4 –7 –6 –5 – 4 –3 –2 –1

4 (2) 6

−2

+2

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

4

5

6

7

−4

–7 –6 –5 – 4 –3 –2 –1

4 2 2

−4 +2 –7 –6 –5 – 4 –3 –2 –1

4 (2) 2

+4 −2 –7 –6 –5 – 4 –3 –2 –1

0

1

2

3

The pattern for the addition of integers shown on the number line can be summarized in the following rule.


Addition of Integers

• Numbers with the same sign To add two numbers with the same sign, add the absolute values of the numbers. Then attach the sign of the addends. • Numbers with different signs To add two numbers with different signs, find the absolute value of each number. Then subtract the smaller of these numbers from the larger. Attach the sign of the number with the larger absolute value.

HOW TO Add: (9) 8 9 9 8 8 • The signs are different. Find the absolute value of each number. 981 • Subtract the smaller number from the larger. (9) 8 1 • Attach the sign of the number with the larger absolute value. Because 9 8, use the sign of 9.


HOW TO

Add: (23) 47 (18) 5

To add more than two numbers, add the first two numbers. Then add the sum to the third number. Continue until all the numbers are added. (23) 47 (18) 5 24 (18) 5 65 11 Look at the two expressions below and note that each expression equals the same number. 835 8 (3) 5

8 minus 3 is 5. 8 plus the opposite of 3 is 5.

This example suggests that to subtract two numbers, we add the opposite of the second number to the first number. first second first number number number

the opposite of the second number

40

60

40

(60)

20

40

60

40

(60)

100

40

(60)

40

60

20

40

(60)

40

60

100

HOW TO

Subtract: 21 (40) '

Change this sign to plus.

'

21 (40) 21 40 19

• Rewrite subtraction as addition of the opposite. Then add.

HOW TO

Subtract: 15 51

Change this sign to plus.

'

15 51 15 (51) 36

• Rewrite subtraction as addition of the opposite. Then add.

Change 51 to the opposite of 51.

HOW TO

Subtract: 12 (21) 15

12 (21) 15 12 21 (15) 9 (15) 6

• Rewrite each subtraction as addition of the opposite. Then add.


Change 40 to the opposite of 40.

'

6


Example 3

Add: (52) (39)

You Try It 3

Solution

The signs are the same. Add the absolute values of the numbers: 52 39 91

Your solution

7

Add: 100 (43)

Attach the sign of the addends: (52) (39) 91 Example 4

Add: 37 (52) (21) (7)

You Try It 4

Add: (51) 42 17 (102)

Solution

37 (52) (21) (7) 15 (21) (7) 36 (7) 43

Your solution

Example 5

Subtract: 11 15

You Try It 5

Solution

11 15 11 (15) 26

Your solution

Example 6

Subtract: 14 18 (21) 4

You Try It 6

Solution

14 18 (21) 4 14 (18) 21 (4) 32 21 (4) 11 (4) 15

Your solution

Subtract: 19 (32)

Subtract: 9 (12) 17 4

Objective D

To multiply or divide integers

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Multiplication is the repeated addition of the same number. The product 3 5 is shown on the number line below. 5

5

5

5 is added 3 times.

          


Solutions on p. S1

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

3 5 5 5 5 15

TA K E N O T E Each number of a product is called a factor. For instance, 3 and 5 are factors of the product 3 5 15.

To indicate multiplication, several different symbols are used. 3 5 15

3 5 15

(3)(5) 15

3(5) 15

(3)5 15

Note that when parentheses are used and there is no arithmetic operation symbol, the operation is multiplication.

8


Now consider the product of a positive and a negative number. 5 is added 3 times.

          

TA K E N O T E 3(5) is 3 times 5.

3(5) (5) (5) (5) 15

• Multiplication is repeated addition.

This suggests that the product of a positive number and a negative number is negative. Here are a few more examples.

To find the product of two negative numbers, look at the pattern at the right. As 5 multiplies a sequence of decreasing integers, the products increase by 5.

(5) 7 35

These numbers decrease by 1.

5(3) 5(2) 5(1) 5(0) 5(1) 5(2) 5(3)

The pattern can be continued by requiring that the product of two negative numbers be positive.

'

(6)5 30

'

4(7) 28

15 10 5 0 5 10 15

These numbers increase by 5.

Multiplication of Integers

• Numbers with the same sign To multiply two numbers with the same sign, multiply the absolute values of the numbers. The product is positive. • Numbers with different signs To multiply two numbers with different signs, multiply the absolute values of the numbers. The product is negative.

HOW TO

Multiply: 2(5)(7)(4)

2(5)(7)(4) 10(7)(4) 70(4) 280

• To multiply more than two numbers, multiply the first two numbers. Then multiply the product by the third number. Continue until all the numbers are multiplied.

For every division problem there is a related multiplication problem. TA K E N O T E 8 4 2

because

Division

428 Related multiplication

This fact and the rules for multiplying integers can be used to illustrate the rules for dividing integers. Note in the following examples that the quotient of two numbers with the same sign is positive. 12 4 because 4 3 12 3

12 4 because 4 (3) 12 3

The next two examples illustrate that the quotient of two numbers with different signs is negative. 12 4 because (4)(3) 12 3

12 4 because (4) 3 12 3


In division, the quotient is the result of dividing the divisor into the dividend.


9

Division of Integers

• Numbers with the same sign To divide two numbers with the same sign, divide the absolute values of the numbers. The quotient is positive. • Numbers with different signs To divide two numbers with different signs, divide the absolute values of the numbers. The quotient is negative.

Simplify:

HOW TO

56 56 7 7

Note that

56 7

(8) 8

12 12 4, 4, 3 3

and

TA K E N O T E The symbol is read “is not equal to.”

If a and b are integers, and b 0, then

12 3

4. This suggests the following rule.

a a a . b b b

Properties of Zero and One in Division

• Zero divided by any number other than zero is zero. 0 0 because 0 a 0 a

For example,

0 0 because 7

0 7 0.

Integrating

Technology

Enter 4 0 on your calculator. You will get an error message.

• Division by zero is not defined. 4 0

To understand that division by zero is not permitted, suppose that were equal to n, where n is some number. Because each division problem has a related multiplication 4 0

problem, n means n · 0 4. But n · 0 4 is impossible because any number times 0 is 0. Therefore, division by 0 is not defined.


• Any number other than zero divided by itself is 1. TA K E N O T E “To the Student” on page xxviii explains how best to use the boxed examples in this text, such as the one below.

a 1, a 0 a

For example,

8 1. 8

For example,

9 9. 1

• Any number divided by one is the number. a a 1

Example 7

Multiply: (3)4(5)

You Try It 7

Solution

(3)4(5) (12)(5) 60

Your solution

Multiply: 8(9)10

Solution on p. S1

10


Example 8

Multiply: 12(4)(3)(5)

You Try It 8

Solution

12(4)(3)(5) (48)(3)(5) 144(5) 720

Your solution

Example 9

Divide: (120) (8)

You Try It 9

Solution

(120) (8) 15

Your solution

95 5

Example 10

Divide:

Solution

95 19 5

Example 11

Divide:

Solution

Multiply: (2)3(8)7

Divide: (135) (9)

Divide:

You Try It 10

72 4

Your solution

81 3

You Try It 11

81 (27) 27 3

Your solution

Divide:

36 12

Solutions on p. S1

Objective E


VIDEO & DVD

CD TUTOR

WEB

SSM

Example 12

You Try It 12

The average temperature on Mercury’s sunlit side is 950°F. The average temperature on Mercury’s dark side is 346°F. Find the difference between these two average temperatures.

The daily low temperatures (in degrees Celsius) during one week were recorded as follows: 6°, 7°, 0°, 5°, 8°, 1°, 1°. Find the average daily low temperature.

Strategy

Your strategy

To find the difference, subtract the average temperature on the dark side (346) from the average temperature on the sunlit side (950). Solution

Your solution

950 (346) 950 346 1296 The difference between these average temperatures is 1296°F. Solution on p. S1


To solve an application problem, first read the problem carefully. The Strategy involves identifying the quantity to be found and planning the steps that are necessary to find that quantity. The Solution involves performing each operation stated in the Strategy and writing the answer.

11


1.1 Exercises Objective A

TA K E N O T E “To the Student” on page xxiii discusses the exercise sets in this textbook.

To use inequality symbols with integers

Place the correct symbol, or , between the two numbers. 1.

8

6

6. 42

2. 14

27

7. 0

16

31

3. 12

1

4.

35

28

8. 17

0

9.

53

46

5. 42

10. 27

19

39

Answer true or false. 11. 13 0

12.

20 3

13.

12 31

14.

9 7

15. 5 2

16. 44 21

17.

4 120

18.

0 8

19.

1 0

20. 10 88

21. Let x {23, 18, 8, 0}. For which values of x is the inequality x 8 a true statement?

22. Let w {33, 24, 10, 0}. For which values of w is the inequality w 10 a true statement?

23. Let a {33, 15, 21, 37}. For which values of a is the inequality a 10 a true statement?

24. Let v {27, 14, 14, 27}. For which values of v is the inequality v 15 a true statement?

25. Let n {23, 1, 0, 4, 29}. For which values of n is the inequality 6 n a true statement?

26. Let m {33, 11, 0, 12, 45}. For which values of m is the inequality 15 m a true statement?


Objective B

To find the additive inverse and absolute value of a number

Find the additive inverse. 28. 8

29. 9

30. 12

31. 28

32. 36

33. (14)

34. (40)

35. (77)

36. (39)

37. (0)

38. (13)

39. 74

40. 96

41. 82

42.

43. 81

44. 38

27. 4

Evaluate.

53

12


Place the correct symbol, or , between the values of the two numbers. 45. 83

58

46. 22

19

47. 43

49. 68

42

50. 12

31

51. 45

52

61

48. 71

92

52. 28

43

53. Let p {19, 0, 28}. Evaluate p for each element of the set.

54. Let q {34, 0, 31}. Evaluate q for each element of the set.

55. Let x {45, 0, 17}. Evaluate x for each element of the set.

56. Let y {91, 0, 48}. Evaluate y for each element of the set.

Objective C

To add or subtract integers

57.

a. Explain the rule for adding two integers with the same sign. b. Explain the rule for adding two integers with different signs.

58.

Explain how to rewrite the subtraction 8 6 as addition of the opposite.

59. 3 (8)

60. 6 (9)

61. 8 3

62. 9 2

63. 3 (80)

64. 12 (1)

65. 23 (23)

66. 12 (12)

67. 16 (16)

68. 17 17

69. 48 (53)

70. 19 (41)

71. 17 (3) 29

72. 13 62 (38)

73. 3 (8) 12

74. 27 (42) (18)

75. 16 8

76. 12 3

77. 7 14

78. 6 9

79. 7 2

80. 9 4

81. 7 (2)

82. 3 (4)

83. 6 (3)

84. 4 (2)

85. 6 (12)

86. 12 16


Add or subtract.


087. 13 (22) 4 (5)

88. 14 (3) 7 (21)

089. 16 (17) (18) 10

90. 25 (31) 24 19

091. 26 (15) (11) (12)

92. 32 40 (8) (19)

093. 14 (15) (11) 40

94. 28 (19) (8) (1)

095. 4 3 2


98. 12 (3) (15)

96.

4 5 12

99.

19 (19) 18

100. 8 (8) 14

97.

12 (7) 8

101. 17 (8) (9)

102.

7 8 (1)

103. 30 (65) 29 4

104. 42 (82) 65 7

105.

16 47 63 12

106. 42 (30) 65 (11)

107. 47 (67) 13 15

108.

18 49 (84) 27

109. 19 17 (36) 12

110. 48 19 29 51

111.

21 (14) 43 12

112. 17 (17) 14 21

Objective D

To multiply or divide integers

113.

Describe the rules for multiplying two integers.

114.

Name the operation in each expression. Justify your answer. a. 87 b. 8 7 c. 8 7 d. xy e. xy

f. x y

13

14


Multiply or divide. 118. 8 7

116. (17)6

121. 11(23)

122. 8(21) 123. (17)14

119. (12)(5) 120. (13)(9)

124. (15)12 125. 6(19)

127. 12 (6)

128. 18 (3)

132. (56) 8

133. (144) 12 134. (93) (3) 135. 48 (8)

137.

49 7

138.

45 5

142.

85 5

143.

120 8

129. (72) (9) 130. (64) (8)

139.

44 4

144.

72 4

140.

36 9

145.

80 5

126. 17(13)

131. 42 6

136. 57 (3)

141.

98 7

146.

114 6

147. 0 (9)

148. 0 (14)

149.

261 9

150.

128 4

151. 9 0

152. (21) 0

153.

132 12

154.

250 25

155.

0 0

156.

58 0

157. 7(5)(3)

158. (3)(2)8

159. 9(7)(4)

160. (2)(6)(4)

161. 16(3)5

162. 20(4)3

163. 4(3)8

164. 5(9)6

165. 3(8)(9)

166. 7(6)(5)

167. (9)7(5)

168. (8)7(10)

169. 7(2)(5)(6)

170. (3)7(2)8

171. 9(4)(8)(10)

172. 11(3)(5)(2)

173. 7(9)(11)4

174. 12(4)7(2)

175. (14)9(11)0

176. (13)(15)(19)0


117. 7 4

115. (14)3


Objective E


Geography The elevation, or height, of places on Earth is measured in relation to sea level, or the average level of the ocean’s surface. The table below shows height above sea level as a positive number and depth below sea level as a negative number. Use the table for Exercises 177 to 179. Continent

Highest Elevation (in meters)

Lowest Elevation (in meters)

Mt. Kilimanjaro

5895

Qattara Depression

−133

Asia

Mt. Everest

8848

Dead Sea

−400

Europe

Mt. Elbrus

5634

Caspian Sea

−28

America

Mt. Aconcagua

6960

Death Valley

−86

Africa

177.

Find the difference in elevation Aconcagua and Death Valley.

between

Mt.

178.

What is the difference in elevation between Mt. Kilimanjaro and the Qattara Depression?

179.

For which continent shown is the difference between the highest and lowest elevations greatest?


Chemistry The table at the right shows the boiling point and melting point in degrees Celsius for three chemical elements. Use this table for Exercises 180 and 181.

Mt. Everest

Chemical Element

Boiling Point

Melting Point −39

Mercury

357

Radon

−62

−71

Xenon

−107

−112

180.

Find the difference between the boiling point and melting point of mercury.

181.

Find the difference between the boiling point and melting point of xenon.

182. Testing To discourage random guessing on a multiple-choice exam, a professor assigns 5 points for a correct answer, 2 points for an incorrect answer, and 0 points for leaving the question blank. What is the score for a student who had 20 correct answers, had 13 incorrect answers, and left 7 questions blank?

183. Testing To discourage random guessing on a multiple-choice exam, a professor assigns 7 points for a correct answer, 3 points for an incorrect answer, and 1 point for leaving the question blank. What is the score for a student who had 17 correct answers, had 8 incorrect answers, and left 2 questions blank?

15

16


The Atmosphere The table at the right shows the average temperatures at different cruising altitudes for airplanes. Use the table for Exercises 184 to 186. 184.

Cruising Altitude

What is the difference between the average temperatures at 12,000 ft and at 40,000 ft?

185.

What is the difference between the average temperatures at 40,000 ft and at 50,000 ft?

186.

How much colder is the average temperature at 30,000 ft than at 20,000 ft?

Average Temperature

12,000 ft

16°F

20,000 ft

−12°F

30,000 ft

−48°F

40,000 ft

−70°F

50,000 ft

−70°F

Meteorology A meteorologist may report a wind-chill temperature. This is the equivalent temperature, including the effects of wind and temperature, that a person would feel in calm air conditions. The table below gives the windchill temperature for various wind speeds and temperatures. For instance, when the temperature is 5°F and the wind is blowing at 15 mph, the wind-chill temperature is 13°F. Use this table for Exercises 187 and 188. Wind Chill Factors Wind Speed (in mph)

Thermometer Reading (in degrees Fahrenheit) 20

15

10

5

0

−5

−10

−15

−20

−25

−30

−35

−40

19

13

7

1

−5

−11

−16

−22

−28

−34

−40

−46

−52

−57

−45 −63

10

15

9

3

−4

−10

−16

−22

−28

−35

−41

−47

−53

−59

−66

−72

15

13

6

0

−7

−13

−19

−26

−32

−39

−45

−51

−58

−64

−71

−77

20

11

4

−2

−9

−15

−22

−29

−35

−42

−48

−55

−61

−68

−74

−81

25

9

3

−4

−11

−17

−24

−31

−37

−44

−51

−58

−64

−71

−78

−84

30

8

1

−5

−12

−19

−26

−33

−39

−46

−53

−60

−67

−73

−80

−87

35

7

0

−7

−14

−21

−27

−34

−41

−48

−55

−62

−69

−76

−82

−89

40

6

−1

−8

−15

−22

−29

−36

−43

−50

−57

−64

−71

−78

−84

−91

45

5

−2

−9

−16

−23

−30

−37

−44

−51

−58

−65

−72

−79

−86

−93

187.

When the thermometer reading is 5°F, what is the difference between the wind-chill factor when the wind is blowing at 10 mph and when the wind is blowing at 25 mph?

188.

When the thermometer reading is 20°F, what is the difference between the wind-chill factor when the wind is blowing at 15 mph and when the wind is blowing at 25 mph?

APPLYING THE CONCEPTS 189.

If 4x equals a positive integer, is x a positive or a negative integer? Explain your answer.

190. Is the difference between two integers always smaller than either of the integers? If not, give an example in which the difference between two integers is greater than either integer.


25 5

Section 1.2 / Rational and Irrational Numbers

1.2 Objective A Point of Interest As early as A.D. 630, the Hindu mathematician Brahmagupta wrote a fraction as one number over another separated by a space. The Arab mathematician al Hassar (around A.D. 1050) was the first to show a fraction with a horizontal bar separating the numerator and denominator.

17

Rational and Irrational Numbers To write a rational number as a decimal

VIDEO & DVD

CD TUTOR

SSM

WEB

A rational number is the quotient of two integers. A rational number written in the following way is commonly called a fraction. Here are some examples of rational numbers. 3 , 4

4 , 9

15 , 4

8 , 1

5 6

Rational Numbers

a A rational number is a number that can be written in the form , where a and b are inteb gers and b 0.

Because an integer can be written as the quotient of the integer and 1, every integer is a rational number. For instance, 8 8 1

6 6 1 Point of Interest Simon Stevin (1548–1620) was the first to name decimal numbers. He wrote the number 2.345 as 2 0 3 1 4 2 5 3. He called the whole number part the commencement; the tenths digit was prime, the hundredths digit was second, the thousandths digit was third, and so on.

A number written in decimal notation is also a rational number.

three-tenths

0.3

3 10

forty-three thousandths

43 1000

A rational number written as a fraction can be rewritten in decimal notation.

HOW TO

Write

5 8

as a decimal.

The fraction bar can be read “”. Copyright © Houghton Mifflin Company. All rights reserved.

0.043

5 58 8 0.625 ' 8 5.000 4.8 20 16 40 40 0 ' 5 0.625 8

This is called a terminating decimal.

Write

4 11

as a decimal.

0.3636 . . . ' This is called a repeating 11 4.0000 decimal. 3.3 70 66 40 33 70 66 4 ' The remainder is never zero.

The remainder is zero.

4 0.36 ' 11

The bar over the digits 3 and 6 is used to show that these digits repeat.

18


Example 1

Write

8 11

You Try It 1

as a decimal. Place a bar over the

Write

repeating digits of the decimal.


repeating digits of the decimal.

8 8 11 0.7272 . . . 0.72 11

Solution

4 9

Your solution Solution on p. S1

Objective B

To convert among percents, fractions, and decimals

VIDEO & DVD

CD TUTOR

WEB

SSM

Percent means “parts of 100.” Thus 27% means 27 parts of 100. In applied problems involving percent, it may be necessary to rewrite a percent as a fraction or decimal or to rewrite a fraction or decimal as a percent. To write a percent as a fraction, remove the percent sign and multiply by

1 . 100

Write 27% as a fraction.

HOW TO

27% 27

1 100

27 100

• Remove the percent sign and multiply by

1 . 100

27% of the region is shaded.

To write a percent as a decimal, remove the percent sign and multiply by 0.01. 33%

33(0.01)

0.33 '

Move the decimal point two places to the left. Then remove the percent sign.

'

A fraction or decimal can be written as a percent by multiplying by 100%. 5 500 1 5 (100%) % 62.5%, or 62 % 8 8 8 2 0.82(100%)

82% '

Move the decimal point two places to the right. Then write the percent sign.

' Example 2

You Try It 2

Write 130% as a fraction and as a decimal.

Write 125% as a fraction and as a decimal.

Solution

Your solution

1 130 13 100 100 10 130% 130(0.01) 1.30 130% 130

Solution on p. S1


0.82


Example 3

Write

5 6

19

You Try It 3

as a percent.

Write

1 3

as a percent.

Solution

5 5 500 1 (100%) % 83 % 6 6 6 3

Your solution

Example 4

Write 0.092 as a percent.

You Try It 4

Solution

0.092 0.092(100%) 9.2%

Your solution


Solutions on p. S1

Objective C

To add or subtract rational numbers

VIDEO & DVD

CD TUTOR

WEB

SSM

Fractions with the same denominator are added by adding the numerators and placing the sum over the common denominator.

Addition of Fractions

To add two fractions with the same denominator, add the numerators and place the sum over the common denominator.

a b ab c c c

To add fractions with different denominators, first rewrite the fractions as equivalent fractions with a common denominator. Then add the fractions. The least common denominator is the least common multiple (LCM) of the denominators. This is the smallest number that is a multiple of each of the denominators.


TA K E N O T E You can find the LCM by multiplying the denominators and then dividing by the common factor of the two denominators. In the case of 6 and 10, 6 10 60. Now divide by 2, the common factor of 6 and 10. 60 2 30

TA K E N O T E The least common multiple of the denominators is frequently called the least common denominator (LCD).

HOW TO

5 3 Add: 6 10 The LCM of 6 and 10 is 30. Rewrite the fractions as equivalent fractions with the denominator 30. Then add the fractions.

5 3 5 5 3 3 25 9 25 9 16 8 6 10 6 5 10 3 30 30 30 30 15

To subtract fractions with the same denominator, subtract the numerators and place the difference over the common denominator. HOW TO

4 7 Subtract: 9 12

Rewrite subtraction as addition of the opposite. The LCM of 9 and 12 is 36. Rewrite the fractions as equivalent fractions with the denominator 36.

7 4 9 12

4 7 16 21 16 21 5 9 12 36 36 36 36

20


To add or subtract decimals, write the numbers so that the decimal points are in a vertical line. Then proceed as in the addition or subtraction of integers. Write the decimal point in the answer directly below the decimal points in the problem. Add: 114.039 84.76 114.039 114.039 • The signs are different. Find the absolute value of each number. 84.76 84.76

HOW TO

114.039 084.76 29.279

• Subtract the smaller of these numbers from the

114.039 84.76 29.279

• Attach the sign of the number with the larger

larger.

absolute value. Because u2114.039u . u84.76u, use the sign of 2114.039. Example 5

You Try It 5

3 1 5 Simplify: 4 6 8

Simplify:

Solution

The LCM of 4, 6, and 8 is 24.

7 5 3 8 6 4

Your solution

3 1 5 18 4 15 4 6 8 24 24 24

18 4 15 24 29 29 24 24

Example 6

Subtract: 42.987 98.61

You Try It 6

Solution

42.987 98.61 42.987 (98.61) 55.623

Your solution

Subtract: 16.127 67.91

Solutions on p. S1

To multiply or divide rational numbers

VIDEO & DVD

The product of two fractions is the product of the numerators divided by the product of the denominators. Multiply:

HOW TO

3 12 3 12 8 17 8 17 1

9 34

WEB

a c ac b d bd

3 12 8 17 • Multiply the numerators. Multiply the denominators.

1

3223 2 2 2 17 1

CD TUTOR

1

• Write the prime factorization of each factor. Divide by the common factors.

• Multiply the factors in the numerator and the factors in the denominator.

SSM


Objective D


21

To divide fractions, invert the divisor. Then multiply the fractions.

TA K E N O T E To invert the divisor means to write its reciprocal. The 18 25 reciprocal of is . 25 18

HOW TO

Divide:

3 18 10 25

The signs are different. The quotient is negative.

3 18 10 25

3 18 10 25 1

1

355 25233 1

3 25 10 18

1

3 25 10 18

5 12

To multiply decimals, multiply as with integers. Write the decimal point in the product so that the number of decimal places in the product equals the sum of the numbers of decimal places in the factors. HOW TO

Multiply: 6.89(0.00035)

6.89 .0.00035 3445 .0.20670 0.0024115

2 decimal places 5 decimal places

• Multiply the absolute values.

7 decimal places

6.89(0.00035) 0.0024115

• The signs are different. The product is negative.

To divide decimals, move the decimal point in the divisor to the right to make the divisor a whole number. Move the decimal point in the dividend the same number of places to the right. Place the decimal point in the quotient directly over the decimal point in the dividend. Then divide as with whole numbers. TA K E N O T E

HOW TO

Divide:

in the divisor and then in the dividend. Place the decimal point in the quotient above the decimal point in the dividend.

Divide:

The quotient is positive.

5 5 8 40

• Move the decimal point 2 places to the right

You Try It 7

5 5 8 40

Solution

'

Example 7

Divide: 1.32 0.27. Round to the nearest tenth.

4.88 4.9 0.27.1.32.00 1.08 2400 2160 240 216 24 '


The symbol is used to indicate that the quotient is an approximate value that has been rounded off.

3 5 8 12

Your solution

5 5 5 40 5 40 8 40 8 5 85 1

1

1

1

52225 5 5 2225 1 1

1

1

1

Solution on p. S2

22


Example 8

Multiply: 4.29(8.2)

You Try It 8

Multiply: 5.44(3.8)

Solution

The product is negative.

Your solution

4.29 .8.2 858 3432 35.178 4.29(8.2) 35.178

Solution on p. S2

Objective E

VIDEO & DVD

To evaluate exponential expressions

CD TUTOR

WEB

SSM

Repeated multiplication of the same factor can be written using an exponent. a a a a a4 ' exponent

'

'

René Descartes (1596–1650) was the first mathematician to use exponential notation extensively as it is used today. However, for some unknown reason, he always used xx for x2.

2 2 2 2 2 25 ' exponent base

base

The exponent indicates how many times the factor, called the base, occurs in the multiplication. The multiplication 2 2 2 2 2 is in factored form. The exponential expression 25 is in exponential form. 21 is read “the first power of 2” or just 2. Usually the exponent 1 is not written. 22 is read “the second power of 2” or “2 squared.” 23 is read “the third power of 2” or “2 cubed.” 24 is read “the fourth power of 2.” a4 is read “the fourth power of a.” There is a geometric interpretation of the first three natural-number powers.

41 = 4 Length: 4 ft

42 = 16 Area: 16 ft2

43 = 64 Volume: 64 ft3


Point of Interest


23

To evaluate an exponential expression, write each factor as many times as indicated by the exponent. Then multiply. Evaluate (2)4.

HOW TO

(2)4 (2)(2)(2)(2) 16

• Write (22) as a factor 4 times. • Multiply.

Evaluate 24.

HOW TO

24 (2 2 2 2) 16

• Write 2 as a factor 4 times. • Multiply.

From these last two examples, note the difference between (2)4 and 24. (2)4 16 24 (24) 16

Example 9

Evaluate 53.

You Try It 9

Solution

53 (5 5 5) 125

Your solution

Example 10

Evaluate (4)4.

You Try It 10

Solution

(4)4 (4)(4)(4)(4) 256

Your solution

Example 11

Evaluate (3)2 23.

You Try It 11

Solution

(3)2 23 (3)(3) (2)(2)(2) 9 8 72

Your solution


Example 12 Evaluate Solution

2 3

2 3

3

2 3

3

.

2 3

2 3

You Try It 12

Evaluate 63.

Evaluate (3)4.

Evaluate (33)(2)3.

Evaluate

2 5

2

.

Your solution

222 8 333 27

Example 13

Evaluate 4(0.7)2.

You Try It 13

Solution

4(0.7)2 4(0.7)(0.7) 2.8(0.7) 1.96

Your solution

Evaluate 3(0.3)3.

Solutions on p. S2

24


Objective F

To simplify numerical radical expressions

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A square root of a positive number x is a number whose square is x. A square root of 16 is 4 because 42 16. A square root of 16 is 4 because (4)2 16.

Square Roots of Perfect Squares 1 1 4 2 9 3 16 4 25 5 36 6 49 7 64 8 81 9 100 10 121 11 144 12

Every positive number has two square roots, one a positive number and one a negative number. The symbol “ ,” called a radical sign, is used to indicate the positive or principal square root of a number. For example, 16 4 and 25 5. The number under the radical sign is called the radicand. When the negative square root of a number is to be found, a negative sign is placed in front of the radical. For example, 16 4 and 25 5. The square of an integer is a perfect square. 49, 81, and 144 are examples of perfect squares.

72 49 92 81 122 144

The principal square root of an integer that is a perfect square is a positive integer.

49 7 81 9 144 12

If a number is not a perfect square, its square root can only be approximated. For example, 2 and 7 are not perfect squares. The square roots of these numbers are irrational numbers. Their decimal representations never terminate or repeat. 2 1.4142135 . . .

7 2.6457513 . . .

Recall that rational numbers are fractions such as

6 7

or

10 , 3

where the numer-

Real Numbers

The rational numbers and the irrational numbers taken together are called the real numbers.

TA K E N O T E Recall that a factor of a number divides the number evenly. For instance, 6 is a factor of 18. The perfect square 9 is also a factor of 18. 9 is a perfect-square factor of 18, whereas 6 is not a perfect-square factor of 18.

Radical expressions that contain radicands that are not perfect squares are frequently written in simplest form. A radical expression is in simplest form when the radicand contains no factor greater than 1 that is a perfect square. For instance, 50 is not in simplest form because 25 is a perfect-square factor of 50. The radical expression 15 is in simplest form because there are no perfectsquare factors of 15 that are greater than 1. The Product Property of Square Roots and a knowledge of perfect squares are used to simplify radicands that are not perfect squares.


ator and denominator are integers. Rational numbers are also represented by repeating decimals, such as 0.25767676 . . . , and by terminating decimals, such as 1.73. An irrational number is neither a repeating nor a terminating decimal. For instance, 2.45445444544445 . . . is an irrational number.


25

The Product Property of Square Roots

If a and b are positive real numbers, then ab a b .

TA K E N O T E From the example at the right, 72 6 2. The two expressions are different representations of the same number. Using a calculator, we find that 72 8.485281 and 6 2 8.485281.

HOW TO

Simplify: 72

72 36 2 36 2 6 2

• Write the radicand as the product of a perfect square and a factor that does not contain a perfect square. • Use the Product Property of Square Roots to write the expression as a product. • Simplify 36.

Note that 72 must be written as the product of a perfect square and a factor that does not contain a perfect square. Therefore, it would not be correct to rewrite 72 as 9 8 and simplify the expression as shown at the right. Although 9 is a perfectsquare factor of 72, 8 has a perfect square factor (8 4 2). Therefore, 8 is not in simplest form. Remember to find the largest perfect-square factor of the radicand.

HOW TO

72 9 8 9 8 3 8 Not in simplest form

Simplify: 16


Because the square of any real number is positive, there is no real number whose square is 16. 16 is not a real number.

Example 14

Simplify: 3 90

You Try It 14

Solution

3 90 3 9 10 3 9 10 3 3 10 9 10

Your solution

Example 15

Simplify: 252

You Try It 15

Solution

252 36 7 36 7 6 7

Your solution

Simplify: 5 32

Simplify: 216

Solutions on p. S2

26


Objective G


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One of the applications of percent is to express a portion of a total as a percent. For instance, a recent survey of 450 mall shoppers found that 270 preferred the mall closest to their home even though it did not have as much store variety as a mall farther from home. The percent of shoppers who preferred the mall closest to home can be found by converting a fraction to a percent. Portion preferring mall closest to home 270 Total number surveyed 450 0.60 60% The Congressional Budget Office projected that the total surpluses for 2001 through 2011 would be $5.6 trillion. The number 5.6 trillion means

        

5.6 1,000,000,000,000 5,600,000,000,000 1 trillion

Numbers such as 5.6 trillion are used in many instances because they are easy to read and offer an approximation of the actual number. The table below shows the net incomes, in millions of dollars, for the third quarter of 2004 for three companies. Note: Negative net income indicates a loss. Use this table for Example 16 and You Try It 16.

Company

Net Income, 3rd Quarter of 2004 (in millions of dollars)

America West

−47.1

FLYi

−82.7 −2.1

Example 16

You Try It 16

If throughout each quarter of 2005, America West’s net income remained at the same level as in the third quarter of 2004, what would be America West’s annual income for 2005?

If throughout each quarter of 2005, Frontier Airlines’s net income remained at the same level as in the third quarter of 2004, what would be Frontier Airlines’s annual net income for 2005?

Strategy

Your strategy

To determine the 2005 annual net income, multiply the net income for the third quarter of 2004 (47.1) by the number of quarters in one year (4). Solution

Your solution

447.1 188.4 The annual net income for America West for 2005 would be $188.4 million. Solution on p. S2


Frontier Airlines


27


To write a rational number as a decimal

Write as a decimal. Place a bar over the repeating digits of a repeating decimal. 1.

1 8

2.

7 8

3.

2 9

4.

8 9

5.

1 6

6.

5 6

7.

9 16

8.

15 16

9.

7 12

10.

11 12

11.

6 25

12.

14 25

13.

9 40

14.

21 40

15.

5 11

Objective B

To convert among percents, fractions, and decimals

16.

a. Explain how to convert a fraction to a percent. b. Explain how to convert a percent to a fraction. c. Explain how to convert a decimal to a percent. d. Explain how to convert a percent to a decimal.

17.

Explain why multiplying a number by 100% does not change the value of the number.


Write as a fraction and as a decimal. 18. 75%

19.

40%

20.

64%

21. 88%

22.

125%

23. 160%

24.

19%

25.

87%

26. 5%

27.

450%

1 30. 12 % 2

1 31. 37 % 2

2 32. 66 % 3

1 35. 6 % 4

1 36. 83 % 3

3 37. 5 % 4

Write as a fraction. 1 28. 11 % 9

33.

1 % 4

2 29. 4 % 7

34.

1 % 2

28


Write as a decimal. 38. 7.3%

39.

9.1%

40.

15.8%

41. 16.7%

42.

0.3%

43. 0.9%

44.

9.9%

45.

9.15%

46. 121.2%

47.

18.23%

48. 0.15

49.

0.37

50.

0.05

51. 0.02

52.

0.175

53. 0.125

54.

1.15

55.

1.36

56. 0.008

57.

0.004

62.

5 11

67.

2

Write as a percent.

58.

27 50

59.

83 100

60.

1 3

61.

63.

4 9

64.

7 8

65.

9 20

66. 1

Objective C

3 8

2 3

1 2

To add or subtract rational numbers

Add or subtract. 5 5 6 9

69.

6 17 13 26

70.

7 5 12 8

71.

5 3 8 4

72.

3 11 5 12

73.

11 5 12 6

74.

2 11 3 18

75.

76.

1 5 2 3 6 9

77.

1 2 1 2 3 6

78.

5 3 7 16 4 8

79.

5 11 8 12

1 3 1 2 8 4


68.


80.

3 7 4 12

7 8

84. 13.092 6.9

81.

1 1 1 3 4 5

82.

2 1 5 3 2 6

83.

85.

2.54 3.6

86.

5.43 7.925

87. 16.92 6.925

1 1 5 16 8 2

88. 3.87 8.546

89.

6.9027 17.692

90.

2.09 6.72 5.4

91. 18.39 4.9 23.7

92.

19 (3.72) 82.75

93.

3.07 (2.97) 17.4

94. 16.4 (3.09) 7.93

95.

3.09 4.6 (27.3)

96.

2.66 (4.66) 8.2

Objective D

To multiply or divide rational numbers

Multiply or divide. 97.


29

1 3 2 4

98.

2 3 9 14

99.

101.

1 8 2 9

102.

5 8 12 15

3 8

4 15

100.

103.

5 7 16 8 12 25

104.

105.

1 3 2 4

106.

3 1 8 4

107.

5 3 6 4

108.

5 15 12 32

110.

7 2 10 5

111.

15 3 64 40

3 4

109.

7 4 8 21

8 27

5 12

8 15

1 3

5 8

30


1 5 8 12

113.

115. 1.2(3.47)

116.

118. (6.9)(4.2)

121. 1.2(0.5)(3.7)

112.

4 2 9 3

4 6 11 9

114.

(0.8)6.2

117.

(1.89)(2.3)

119.

1.06(3.8)

120.

2.7(3.5)

122.

2.4(6.1)(0.9)

123.

2.3(0.6)(0.8)

Divide. Round to the nearest hundredth. 124. 1.27 (1.7)

125.

9.07 (3.5)

126.

0.0976 0.042

127. 6.904 1.35

128.

7.894 (2.06)

129.

354.2086 0.1719

Objective E

To evaluate exponential expressions

Evaluate. 130. 62

131. 74

132. 72

133. 43

135. (2)3

136. (3)4

137. (5)3

138.

140. (0.3)2

141. (1.5)3

142.

143.

145. (2) (2)2

146. 23 33 (4)

149. (2) 23 (3)2

150.

2 3

2

1 33 4

2 3

2

33

2

1 2

139.

3

147. (3)3 52 10

151.

3 4

2

(4) 23

8

3 4

3

144. (0.3)3 23

148. (7) 42 32

152. 82 (3)5 5


1 2

134. (3)2


Objective F

31

To simplify numerical radical expressions

Simplify. 153. 16

154. 64

155. 49

156. 144

157. 32

158. 50

159. 8

160. 12

161. 6 18

162. 3 48

163. 5 40

164. 2 28

165. 15

166. 21

167. 29

168. 13

169. 9 72

170. 11 80

171. 45

172. 225

173. 0

174. 210

175. 6 128

176. 9 288

Find the decimal approximation rounded to the nearest thousandth. 177. 240

178. 300

179. 288

180. 600

181. 256

182. 324

183. 275

184. 450

185. 245

186. 525

187. 352

188. 363

Objective G


189.


Business The table below shows the annual net incomes for the periods ending in January 2004 and January 2003 for two companies. Figures are in millions of dollars. Profits are shown as positive numbers; losses are shown as negative numbers. Round to the nearest thousand dollars. a. What was the average monthly net income for TiVo, Inc., for the period ending in January 2004? b. Find the difference between the annual net income for Warnaco for the period ending January 2004 and the period ending January 2003.

Company

TiVo, Inc. Warnaco Group, Inc.

Annual Net Income for Period Ending January 2004 (in millions of dollars)

Annual Net Income for Period Ending January 2003 (in millions of dollars)

−32.018

−80.596

2,360.423

−964.863

32


190.

191.

The Stock Market At the close of the stock markets on December 2, 2004, the indexes were posted as shown below, along with the increase or decrease, shown as a negative number for that day. At what level were the indexes at the close of the day on December 1, 2004? Index

Points at Market Close

Decrease for the Day

Dow Jones Industrial Average

10,585.12

−5.10

Standard & Poor's 500

1,190.33

−1.04

NASDAQ

2,143.57

+5.34

Halloween Spending In a recent year, the average consumer spent approximately $44 on Halloween merchandise. The breakdown is shown in the graph at the right (Source: BIGresearch for the National Retail Federation). What percent of the total is the amount spent on decorations?

Greeting cards $3

Decorations $11 Costumes $15

192.

The Federal Budget The table at the right shows the surplus or deficit, in billions of dollars, for the federal budget every fifth year from 1945 to 1995 and every year from 1995 to 2000 (Source: U.S. Office of Management and Budget). The negative sign () indicates a deficit. a. Find the difference between the deficits in the years 1980 and 1985. b. Calculate the difference between the surplus in 1960 and the deficit in 1955. c. How many times greater was the deficit in 1985 than in 1975? Round to the nearest whole number. d. What was the average deficit, in millions of dollars, per quarter for the year 1970? e. Find the average surplus or deficit for the years 1995 through 2000. Round to the nearest million.

APPLYING THE CONCEPTS

Year

Federal Budget Surplus or Deficit

1945

−47.533

1950

−3.119

1955

−2.993

1960

0.301

1965

−1.411

1970

−2.842

1975

−53.242

1980

−73.835

1985

−212.334

1990

−221.194

1995

−163.899

1996

−107.450

1997

−21.940

1998

69.246

1999

79.263

2000

117.305

193. List the whole numbers between 8 and 90.

194. Use a calculator to determine the decimal representations of 73 . 99

Make a conjecture as to the decimal representation of

conjecture work for

195.

33 ? 99

What about

83 . 99

17 45 , , 99 99

and

Does your

1 ? 99

Describe in your own words how to simplify a radical expression.


Candy $15

Section 1.3 / The Order of Operations Agreement

1.3 Objective A

33

The Order of Operations Agreement To use the Order of Operations Agreement to simplify expressions

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Let’s evaluate 2 3 5. There are two arithmetic operations, addition and multiplication, in this expression. The operations could be performed in different orders.

235

Add first.

Then add.

2 15

Then multiply.

235

  

  

Multiply first.

    

  

55

17

25

In order to prevent there being more than one answer when simplifying a numerical expression, an Order of Operations Agreement has been established.

The Order of Operations Agreement

Perform operations inside grouping symbols. Grouping symbols include parentheses ( ), brackets [ ], braces { }, absolute value symbols , and the fraction bar. Step 2 Simplify exponential expressions. Step 3 Do multiplication and division as they occur from left to right. Step 4 Do addition and subtraction as they occur from left to right. Step 1

HOW TO

Evaluate 12 24(8 5) 22.


12 24(8 5) 22 12 24(3) 22

• Perform operations inside grouping symbols.

12 24(3) 4

• Simplify exponential expressions.

12 72 4

• Do multiplication and division as they occur from left to right.

12 18 6

• Do addition and subtraction as they occur from left to right.

One or more of the above steps may not be needed to evaluate an expression. In that case, proceed to the next step in the Order of Operations Agreement.

34


When an expression has grouping symbols inside grouping symbols, perform the operations inside the inner grouping symbols first.

Study

Tip

HOW TO

The HOW TO feature gives an example with explanatory remarks. Using paper and pencil, you should work through the example. See AIM for Success, pages xxvii –xxviii.

Evaluate 6 [4 (6 8)] 22.

6 [4 (6 8)] 22 6 [4 (2)] 22

• Perform operations inside grouping symbols.

6 6 22 664

• Simplify exponential

14

• Do multiplication and division

expressions. as they occur from left to right.

5

• Do addition and subtraction as they occur from left to right.

Example 1

You Try It 1

Evaluate 4 3[4 2(6 3)] 2.

Evaluate 18 5[8 2(2 5)] 10.

Solution

Your solution

4 3[4 2(6 3)] 2 4 3[4 2 3] 2 4 3[4 6] 2 4 3[2] 2 462 43 7 Example 2

You Try It 2

Evaluate 27 (5

2)2

(3)2

4.

Your solution

27 (5 2)2 (3)2 4 27 32 (3)2 4 27 9 9 4 394 3 36 39 Example 3

Evaluate (1.75

You Try It 3

1.3)2

0.025 6.1.

Solution

Evaluate (6.97 4.72)2 4.5 0.05. Your solution

(1.75 1.3)2 0.025 6.1 (0.45)2 0.025 6.1 0.2025 0.025 6.1 8.1 6.1 14.2 Solutions on p. S2


Solution

Evaluate 36 (8 5)2 (3)2 2.

Section 1.3 / The Order of Operations Agreement


To use the Order of Operations Agreement to simplify expressions

01.

Why do we need an Order of Operations Agreement?

02.

Describe each step in the Order of Operations Agreement.


Evaluate by using the Order of Operations Agreement. 03. 4 8 2

04.

22 3 3

05. 2(3 4) (3)2

06. 16 32 23

07.

24 18 3 2

08. 8 (3)2 (2)

09. 8 2(3)2

10.

16 16 2 4

11. 12 16 4 2

12. 16 2 42

13.

27 18 (32)

14. 4 12 3 2

15. 16 15 (5) 2

16.

14 22 (4 7)

17. 14 22 4 7

18. 10 5 8 23

19.

3 2[8 (3 2)]

20. 22 4[16 (3 5)]

22.

24

24. 96 2[12 (6 2)] 32

25.

4[16 (7 1)] 10

26. 18 2 42 (3)2

27. 20 (10 23) (5)

28.

16 3(8 3)2 5

29. 4(8) [2(7 3)2]

21. 6

16 4 2 22 2

32 (5) 85

23. 18 9 23 (3)

35

36

30.


(10) (2) 2 4 62 30

33. 0.3(1.7 4.8) (1.2)2

33 7 (2)2 23 2

31.

16 4

34.

(1.8)2 2.52 1.8

32. (0.2)2 (0.5) 1.72

35. (1.65 1.05)2 0.4 0.8

APPLYING THE CONCEPTS 36. Find two fractions between this question.)

2 3

3 4

and . (There is more than one answer to

37. A magic square is one in which the numbers in every row, column, and diagonal sum to the same number. Complete the magic square at the right.

2 3 1 6

38. For each part below, find a rational number r that satisfies the condition. a. r 2 r b. r 2 r c. r 2 r

5 6

1 3

39. Electric Vehicles In a survey of consumers, 9% said they would buy an electric vehicle. Approximately 43% said they would be willing to pay $1500 more for a new car if the car had an EPA rating of 80 mpg. If your car now gets 28 mpg and you drive approximately 10,000 mi per year, in how many months would your savings on gasoline pay for the increased cost of such a car? Assume the average cost for gasoline is $2.00 per gallon. Round to the nearest whole number.

40. Find three different natural numbers a, b, and c such that

1 1 1 a b c

is a

41.

The following was offered as the simplification of 6 2(4 9). 6 2(4 9) 6 2(5) 8(5) 40 Is this a correct simplification? Explain your answer.

42.

The following was offered as the simplification of 2 33. 2 33 63 216 Is this is a correct simplification? Explain your answer.


natural number.

Section 1.4 / Variable Expressions

Point of Interest Historical manuscripts indicate that mathematics is at least 4000 years old. Yet it was only 400 years ago that mathematicians started using variables to stand for numbers. The idea that a letter can stand for some number was a critical turning point in mathematics.

To evaluate a variable expression

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Often we discuss a quantity without knowing its exact value—for example, the price of gold next month, the cost of a new automobile next year, or the tuition cost for next semester. Recall that a letter of the alphabet can be used to stand for a quantity that is unknown or that can change, or vary. Such a letter is called a variable. An expression that contains one or more variables is called a variable expression. A variable expression is shown at the right. The expression can be rewritten by writing subtraction as the addition of the opposite.

3x2 (5y) 2xy (x) (7)

5 terms

3x2 5y

2xy

x

7

          

    

Note that the expression has five addends. The terms of a variable expression are the addends of the expression. The expression has five terms.

3x2 5y 2xy x 7

    

Objective A

Variable Expressions

                 

1.4

37

variable terms

constant term

The terms 3x2, 5y, 2xy, and x are variable terms. The term 7 is a constant term, or simply a constant. Each variable term is composed of a numerical coefficient and a variable part (the variable or variables and their exponents).

numerical coefficient

3x2 5y

2xy 1x 7

variable part


When the numerical coefficient is 1 or 1, the 1 is usually not written (x 1x and x 1x). Replacing each variable by its value and then simplifying the resulting numerical expression is called evaluating the variable expression.

Integrating

Technology See the appendix Keystroke Guide for instructions on using a graphing calculator to evaluate variable expressions.

HOW TO

Evaluate ab b2 when a 2 and b 3.

ab b2 2(3) (3)2 2(3) 9

• Replace each variable in the expression by its value. • Use the Order of Operations Agreement to simplify the resulting numerical expression.

6 9 15

38


Example 1

You Try It 1

a b when a 3 and b 4. ab 2

Evaluate

Solution

2

a2 b2 ab

Evaluate

a2 b2 when a 5 and b 3. ab

Your solution

32 (4)2 9 16 3 (4) 3 (4)

7 1 7

Example 2

Evaluate x2 3(x y) z2 when x 2, y 1, and z 3.

You Try It 2

Solution

x2 3(x y) z2

Your solution

Evaluate x3 2(x y) z2 when x 2, y 4, and z 3.

22 3[2 (1)] 32 22 3(3) 32 4 3(3) 9 499 5 9 14 Solutions on p. S2

To simplify a variable expression using the Properties of Addition

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Like terms of a variable expression are terms with the same variable part. (Because x2 x x, x2 and x are not like terms.)

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like terms

3x 4 7x 9 x2

Constant terms are like terms. 4 and 9 are like terms.

like terms

To simplify a variable expression, use the Distributive Property to combine like terms by adding the numerical coefficients. The variable part remains unchanged.

Distributive Property

If a, b, and c are real numbers, then a(b c) ab ac.

The Distributive Property can also be written as ba ca (b c)a. This form is used to simplify a variable expression.


Objective B


HOW TO

39

Simplify: 2x 3x

Use the Distributive Property to add the numerical coefficients of the like variable terms. This is called combining like terms. 2x 3x (2 3)x 5x HOW TO

• Use the Distributive Property.

Simplify: 5y 11y • Use the Distributive Property. This step is

5y 11y (5 11)y

usually done mentally.

6y TA K E N O T E Simplifying an expression means combining like terms. A constant term (5) and a variable term (7p) are not like terms and therefore cannot be combined.

HOW TO

Simplify: 5 7p

The terms 5 and 7p are not like terms. The expression 5 7p is in simplest form. In simplifying variable expressions, the following Properties of Addition are used. The Associative Property of Addition

If a, b, and c are real numbers, then (a b) c a (b c).

When three or more like terms are added, the terms can be grouped (with parentheses, for example) in any order. The sum is the same. For example, (3x 5x) 9x 3x (5x 9x) 8x 9x 3x 14x 17x 17x The Commutative Property of Addition


If a and b are real numbers, then a b b a.

When two like terms are added, the terms can be added in either order. The sum is the same. For example, 2x (4x) 4x 2x 2x 2x The Addition Property of Zero

If a is a real number, then a 0 0 a a.

The sum of a term and zero is the term. For example, 5x 0 0 5x 5x

40


The Inverse Property of Addition

If a is a real number, then a (a) (a) a 0.

The sum of a term and its opposite is zero. The opposite of a number is called its additive inverse. 7x (7x) 7x 7x 0 HOW TO

Simplify: 8x 4y 8x y

Use the Commutative and Associative Properties of Addition to rearrange and group like terms. Then combine like terms. 8x 4y 8x y (8x 8x) (4y y)

• This step is usually done mentally.

0 5y 5y HOW TO

Simplify: 4x2 5x 6x2 2x 1

Use the Commutative and Associative Properties of Addition to rearrange and group like terms. Then combine like terms. 4x2 5x 6x2 2x 1 (4x2 6x2) (5x 2x) 1 2x2 3x 1

Example 3

Simplify: 3x 4y 10x 7y

You Try It 3

Solution

3x 4y 10x 7y 7x 11y

Your solution

Example 4

Simplify: x2 7 4x2 16

You Try It 4

Solution

x2 7 4x2 16 5x2 23

Your solution

Simplify: 3a 2b 5a 6b

Simplify: 3y2 7 8y2 14

Objective C

To simplify a variable expression using the Properties of Multiplication

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In simplifying variable expressions, the following Properties of Multiplication are used. The Associative Property of Multiplication

If a, b, and c are real numbers, then (a b) c a (b c).

When three or more factors are multiplied, the factors can be grouped in any order. The product is the same. For example, 2(3x) (2 3)x 6x


Solutions on p. S2


Study

Tip

Some students think that they can “coast” at the beginning of this course because they have been taught this material before. However, this chapter lays the foundation for the entire course. Be sure you know and understand all the concepts presented. For example, study the properties of multiplication presented in this lesson.

41

The Commutative Property of Multiplication

If a and b are real numbers, then a b b a.

Two factors can be multiplied in either order. The product is the same. For example, (2x) 3 3 (2x) 6x

The Multiplication Property of One

If a is a real number, then a 1 1 a a.

The product of a term and one is the term. For example, (8x)(1) (1)(8x) 8x

The Inverse Property of Multiplication

If a is a real number, and a is not equal to zero, then 1 1 a a 1. a a

1 a

is called the reciprocal of a.

1 a

is also called the multiplicative inverse of a.

The product of a number and its reciprocal is one. For example,

7

1 1 71 7 7

The multiplication properties just discussed are used to simplify variable expressions.


HOW TO

Simplify: 2(x)

2(x) 2(1 x) [2(1)]x 2x

HOW TO

Simplify:

• Use the Associative Property of Multiplication to group factors.

3 2x 2 3

Use the Associative Property of Multiplication to group factors.

3 2x 2 3

3 2 x 2 3 3 2 x 2 3

1x x

• Note that

2x 2 5 x. 3 3

• The steps in the dashed box are usually done mentally.

42


HOW TO

Simplify: (16x)2

Use the Commutative and Associative Properties of Multiplication to rearrange and group factors. (16x)2 2(16x)

• The steps in the dashed box are usually done mentally.

(2 16)x 32x

Example 5

Simplify: 2(3x2)

You Try It 5

Simplify: 5(4y2)

Solution

2(3x2) 6x2

Your solution

Example 6

Simplify: 5(10x)

You Try It 6

Solution

5(10x) 50x

Your solution

Example 7

Simplify: (6x)(4)

You Try It 7

Solution

(6x)(4) 24x

Your solution

Simplify: 7(2a)

Simplify: (5x)(2)

Solutions on p. S2

To simplify a variable expression using the Distributive Property

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Recall that the Distributive Property states that if a, b, and c are real numbers, then a(b c) ab ac The Distributive Property is used to remove parentheses from a variable expression.

HOW TO

Simplify: 3(2x 7)

3(2x 7) 3(2x) 3(7)


6x 21

HOW TO

Do this step mentally.

Simplify: 5(4x 6)

5(4x 6) 5(4x) (5) 6 20x 30

• Use the Distributive Property. Do this step mentally.


Objective D


HOW TO

43

Simplify: (2x 4)

(2x 4) 1(2x 4) 1(2x) (1)(4)

• Use the Distributive Property. Do these steps mentally.

2x 4 Note: When a negative sign immediately precedes the parentheses, the sign of each term inside the parentheses is changed.

HOW TO

1 Simplify: (8x 12y) 2

1 1 1 (8x 12y) (8x) 2 2 2


(12y)

Do this step mentally.

4x 6y

HOW TO

Simplify: 4(x y) 2(3x 6y)

4(x y) 2(3x 6y) 4x 4y 6x 12y

• Use the Distributive Property twice.

Study

Tip


One of the key instructional features of this text is the Example/You Try It pairs. Each Example is completely worked. You are to solve the You Try It problems. When you are ready, check your solution against the one in the Solutions section. The solutions for You Try It 8 and 9 below are on pages S2–S3 (see the reference at the bottom right of the You Try It box). See AIM for Success, page xxviii.

10x 16y

• Combine like terms.

The Distributive Property is used when an expression inside parentheses contains more than two terms. See the example below. HOW TO

Simplify: 3(4x 2y z)

3(4x 2y z) 3(4x) 3(2y) 3(z) 12x 6y 3z

Example 8

Simplify: 3(5a 7b)

You Try It 8

Solution

3(5a 7b) 15a 21b

Your solution

Example 9

Simplify: (2x 6)2

You Try It 9

Solution

(2x 6)2 4x 12

Your solution

• Use the Distributive Property. Do this step mentally.

Simplify: 8(2a 7b)

Simplify: (3a 1)5

Solutions on pp. S2–S3


Example 10

Simplify: 3(x2 x 5)

You Try It 10

Solution

3(x2 x 5) 3x2 3x 15

Your solution

Example 11

Simplify: 2x 3(2x 7y)

You Try It 11

Solution

2x 3(2x 7y) 2x 6x 21y 4x 21y

Your solution

Example 12

Simplify: 7(x 2y) (x 2y)

You Try It 12

Solution

7(x 2y) (x 2y) 7x 14y x 2y 8x 12y

Your solution

Example 13

Simplify: 2x 3[2x 3(x 7)]

You Try It 13

Solution

2x 3[2x 3(x 7)] 2x 3[2x 3x 21] 2x 3[x 21] 2x 3x 63 5x 63

Your solution

Simplify: 2(x2 x 7)

Simplify: 3y 2(y 7x)

Simplify: 2(x 2y) (x 3y)

Simplify: 3y 2[x 4(2 3y)]

Solutions on p. S3

Objective E

To translate a verbal expression into a variable expression

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One of the major skills required in applied mathematics is the ability to translate a verbal expression into a variable expression. This requires recognizing the verbal phrases that translate into mathematical operations. A partial list of the verbal phrases used to indicate the different mathematical operations follows. Addition

added to more than the sum of increased by the total of

6 added to y 8 more than x the sum of x and z t increased by 9 the total of 5 and y

y6 x8 xz t9 5y


44

45


Subtraction Point of Interest The way in which expressions are symbolized has changed over time. Here is how some of the expressions shown at the right may have appeared in the early 16th century. R p. 8 for x 8. The symbol R was used for a variable to the first power. The symbol p. was used for plus. R m. 2 for x 2. The symbol R is again used for the variable. The symbol m. was used for minus. The square of a variable was designated by Q and the cube of a variable was designated by C. The expression x 3 x 2 was written C p. Q.

Multiplication

Division

Power

HOW TO

minus less than decreased by the difference between

x minus 2 7 less than t m decreased by 3 the difference between y and 4

x2 t7 m3

times

10 times t

of

one-half of x

the product of multiplied by twice

the product of y and z y multiplied by 11 twice d

10t 1 x 2 yz 11y 2d

divided by

x divided by 12

the quotient of

the quotient of y and z

the ratio of

the ratio of t to 9

the square of the cube of

the square of x the cube of a

y4

x 12 y z t 9

x2 a3

Translate “14 less than the cube of x” into a variable expression.

14 less than the cube of x

• Identify the words that indicate

x3 14

• Use the identified operations to

the mathematical operations.


write the variable expression.

In most applications that involve translating phrases into variable expressions, the variable to be used is not given. To translate these phrases, a variable must be assigned to an unknown quantity before the variable expression can be written.

HOW TO Translate “the sum of two consecutive integers” into a variable expression. Then simplify. the first integer: n

• Assign a variable to one of

the next consecutive integer: n 1

• Use the assigned variable to

n (n 1)

write an expression for any other unknown quantity. • Use the assigned variable to write the variable expression. • Simplify the variable expression.

the unknown quantities.

(n n) 1 2n 1

46


Many of the applications of mathematics require that you identify an unknown quantity, assign a variable to that quantity, and then attempt to express another unknown quantity in terms of the variable. (c + 3) lb

HOW TO A confectioner makes a mixture of candy that contains 3 lb more of milk chocolate than of caramel. Express the amount of milk chocolate in the mixture in terms of the amount of caramel in the mixture.

c lb

Amount of caramel in the mixture: c

• Assign a variable to the amount

Amount of milk chocolate in the mixture: c 3

• Express the amount of milk chocolate

of caramel in the mixture. in the mixture in terms of c.

Example 14

You Try It 14

Translate “four times the sum of half of a number and fourteen” into a variable expression. Then simplify.

Translate “five times the difference between a number and sixty” into a variable expression. Then simplify.

Solution

Your solution

the unknown number: n 1 2

half of the number: n the sum of half of the number and 1 2

fourteen: n 14

4

1 n 14 2

Example 15

You Try It 15

The length of a swimming pool is 4 ft less than two times the width. Express the length of the pool in terms of the width.

The speed of a new printer is twice the speed of an older model. Express the speed of the new model in terms of the speed of the older model.

Solution

Your solution

the width of the pool: w the length is 4 ft less than two times the width: 2w 4 Example 16

You Try It 16

A banker divided $5000 between two accounts, one paying 10% annual interest and the second paying 8% annual interest. Express the amount invested in the 10% account in terms of the amount invested in the 8% account.

A guitar string 6 ft long was cut into two pieces. Express the length of the shorter piece in terms of the length of the longer piece.

Solution

Your solution

the amount invested at 8%: x the amount invested at 10%: 5000 x Solutions on p. S3


2n 56


47


To evaluate a variable expression

Evaluate the variable expression when a 2, b 3, and c 4. 1. 6b (a)

2.

bc (2a)

3.

b2 4ac

4. a2 b2

5.

b2 c2

6.

(a b)2

7. a2 b2

8.

2a (c a)2

9.

(b a)2 4c

ac 8

11.

10. b2

5ab 3cb 6

12. (b 2a)2 bc


Evaluate the variable expression when a 2, b 4, c 1, and d 3. 13.

bc d

14.

db c

15.

2d b a

16.

b 2d b

17.

bd ca

18.

2c d ad

19. (b d)2 4a

20. (d a)2 3c

22. 3(b a) bc

23.

b 2a bc2 d

24.

b2 a ad 3c

26.

5 4 a c2 8

27.

4bc 2a b

25.

1 2 3 2 d b 3 8

21. (d a)2 5

3 1 28. b (ac bd) 4 2

2 1 29. d (bd ac) 3 5

30. (b a)2 (d c)2

31. (b c)2 (a d)2

32. 4ac (2a)2

33. 3dc (4c)2

48


Objective B

To simplify a variable expression using the Properties of Addition

Simplify. 34. 6x 8x

35. 12x 13x

36. 9a 4a

37. 12a 3a

38. 4y (10y)

39. 8y (6y)

40. 3b 7

41. 12y 3

42. 12a 17a

43. 3a 12a

44. 5ab 7ab

45. 9ab 3ab

46. 12xy 17xy

47. 15xy 3xy

48. 3ab 3ab

49. 7ab 7ab

1 1 50. x x 2 3

2 3 y 51. y 5 10

52.

54. 3x 5x 3x

55. 8x 5x 7x

56. 5a 3a 5a

53.

2 2 4 2 y y 3 9

57. 10a 17a 3a

58. 5x2 12x2 3x2

59. y2 8y2 7y2

60. 7x (8x) 3y

61. 8y (10x) 8x

62. 7x 3y 10x

63. 8y 8x 8y

64. 3a (7b) 5a b

65. 5b 7a 7b 12a

66. 3x (8y) 10x 4x

67. 3y (12x) 7y 2y

68. x2 7x (5x2) 5x

69. 3x2 5x 10x2 10x


5 2 3 2 x x 8 12

49


Objective C

To simplify a variable expression using the Properties of Multiplication

Simplify. 070. 4(3x)

071. 12(5x)

072. 3(7a)

073. 2(5a)

074. 2(3y)

075. 5(6y)

076. (4x)2

077. (6x)12

078. (3a)(2)

079. (7a)(4)

080. (3b)(4)

081. (12b)(9)

082. 5(3x2)

083. 8(7x2)

084.

1 088. (2x) 2

1 089. (4a) 4

085.

1 (6x2) 6

086.

1 090. (7n) 7

2 100. (12a2) 3


105. (33y)

1 11

Objective D

087.

1 091. (9b) 9

095. (10n)

1 (5a) 5

1 10

096.

1 (9x) 3

106. (6x)

092. (3x)

097.

5 101. (24a2) 8

1 3

1 (8x) 8

1 3

1 (14x) 7

1 102. (16y) 2

107. (10x)

1 5

093. (12x)

1 12

1 (3x2) 3

094. (6y)

1 098. (10x) 5

1 099. (16x) 8

3 103. (8y) 4

104. (16y)

108. (8a)

3 4

1 6

1 4

109. (21y)

To simplify a variable expression using the Distributive Property

Simplify. 110. (x 2)

111. (x 7)

112. 2(4x 3)

113. 5(2x 7)

114. 2(a 7)

115. 5(a 16)

116. 3(2y 8)

117. 5(3y 7)

3 7


1 (6 15y) 3

1 (8x 4y) 2

118. (5 3b)7

119. (10 7b)2

120.

122. 3(5x2 2x)

123. 6(3x2 2x)

124. 2(y 9)

125. 5(2x 7)

126. (3x 6)5

127. (2x 7)7

128. 2(3x2 14)

129. 5(6x2 3)

130. 3(2y2 7)

131. 8(3y2 12)

132. 3(x2 y2)

133. 5(x2 y2)

121.

2 134. (6x 18y) 3

1 135. (x 4y) 2

136. (6a2 7b2)

137. 3(x2 2x 6)

138. 4(x2 3x 5)

139. 2(y2 2y 4)

1 141. (6x 9y 1) 3

142. 4(3a2 5a 7)

143. 5(2x2 3x 7)

144. 3(4x2 3x 4)

145. 3(2x2 xy 3y2)

146. 5(2x2 4xy y2)

147. (3a2 5a 4)

148. (8b2 6b 9)

149. 4x 2(3x 8)

150. 6a (5a 7)

151. 9 3(4y 6)

152. 10 (11x 3)

153. 5n (7 2n)

154. 8 (12 4y)

140.

1 (2x 6y 8) 2


50


51

155. 3(x 2) 5(x 7)

156. 2(x 4) 4(x 2)

157. 12(y 2) 3(7 3y)

158. 6(2y 7) (3 2y)

159. 3(a b) (a b)

160. 2(a 2b) (a 3b)

161. 4[x 2(x 3)]

162. 2[x 2(x 7)]

163. 2[3x 2(4 x)]

164. 5[2x 3(5 x)]

165. 3[2x (x 7)]

166. 2[3x (5x 2)]

167. 2x 3[x (4 x)]

168. 7x 3[x (3 2x)]

169. 5x 2[2x 4(x 7)] 6

Objective E

To translate a verbal expression into a variable expression


Translate into a variable expression. Then simplify. 170. twelve minus a number

171. a number divided by eighteen

172. two-thirds of a number

173. twenty more than a number

174. the quotient of twice a number and nine

175. ten times the difference between a number and fifty

176. eight less than the product of eleven and a number

177. the sum of five-eighths of a number and six

178. nine less than the total of a number and two

179. the difference between a number and three more than the number


180. the quotient of seven and the total of five and a number

181. four times the sum of a number and nineteen

182.

five increased by one-half of the sum of a number and three

183. the quotient of fifteen and the sum of a number and twelve

184.

a number added to the difference between twice the number and four

185. the product of two-thirds and the sum of a number and seven

186.

the product of five less than a number and seven

187. the difference between forty and the quotient of a number and twenty

188.

the quotient of five more than twice a number and the number

189. the sum of the square of a number and twice the number

190.

a number decreased by the difference between three times the number and eight

191. the sum of eight more than a number and one-third of the number

192.

a number added to the product of three and the number

193. a number increased by the total of the number and nine

194.

five more than the sum of a number and six

195. a number decreased by the difference between eight and the number

196.

a number minus the sum of the number and ten

197. the difference between one-third of a number and five-eighths of the number

198.

the sum of one-sixth of a number and fourninths of the number

199. two more than the total of a number and five

200.

the sum of a number divided by three and the number

201. twice the sum of six times a number and seven


52

53


202.

Planets The planet Saturn has 9 more moons than Jupiter (Source: NASA). Express the number of moons Saturn has in terms of the number of moons Jupiter has.

203.

The Olympics The number of nations participating in the Olympic Games in 2004 was 1990 more than the number of nations participating in the Olympic Games in 1896 (Source: USA Today research). Express the number of nations participating in the Olympic Games in 2004 in terms of the number of nations participating in the Olympic Games in 1896. ?

204. Sailing A halyard 12 ft long was cut into two pieces of different lengths. Use one variable to express the lengths of the two pieces.

205. Natural Resources Twenty gallons of crude oil was poured into two containers of different sizes. Use one variable to express the amount of oil poured into each container.


Agriculture In a recent year, Alabama produced one-half the number of pounds of pecans that Texas produced that same year (Source: National Agricultural Statistics Service). Express the amount of pecans produced in Alabama in terms of the amount produced in Texas.

208.

Internal Revenue Service According to the Internal Revenue Service, it takes about one-fifth as much time to fill out Schedule B (interest and dividends) as to fill out Schedule A (itemized deductions). Express the amount of time it takes to fill out Schedule B in terms of the time it takes to fill out Schedule A.

209.

Sports Equipment The diameter of a basketball is approximately 4 times the diameter of a baseball. Express the diameter of a basketball in terms of the diameter of a baseball.

210.

World Population According to the U.S. Bureau of the Census, the world population in the year 2050 is expected to be twice the world population in 1980. Express the world population in 2050 in terms of the world population in 1980.

x

207.

200 mi

206. Travel Two cars start at the same place and travel at different rates in opposite directions. Two hours later the cars are 200 mi apart. Express the distance traveled by the faster car in terms of the distance traveled by the slower car.

S

54


APPLYING THE CONCEPTS 211. Does every number have an additive inverse? If not, which real numbers do not have an additive inverse? 212. Does every number have a multiplicative inverse? If not, which real numbers do not have a multiplicative inverse? 213. Chemistry The chemical formula for glucose (sugar) is C6H12O6. This formula means that there are twelve hydrogen atoms, six carbon atoms, and six oxygen atoms in each molecule of glucose. If x represents the number of atoms of oxygen in a pound of sugar, express the number of hydrogen atoms in the pound of sugar in terms of x. 214. Determine whether the statement is true or false. If the statement is false, give an example that illustrates that it is false. a. Division is a commutative operation. b. Division is an associative operation. c. Subtraction is an associative operation. d. Subtraction is a commutative operation. e. Addition is a commutative operation.

H

O C

H

C

OH

HO

C

H

H

C

OH

H

C

OH

CH 2 OH

215. Metalwork A wire whose length is given as x inches is bent into a square. Express the length of a side of the square in terms of x. x ?

216. For each of the following, determine the first natural number x greater than 2 for which the second expression is larger than the first. a. x3, 3x

b. x4, 4x

c. x5, 5x

d. x6, 6x

3 ft

218.

Give examples of two operations that occur in everyday experience that are not commutative (for example, putting on socks and then shoes).

219.

Choose any number a. Evaluate the expressions 6a2 2a 10 and 2a(3a 4) 10(a 1). Now choose a different number and evaluate the expressions again. Repeat this two more times with different numbers. What conclusions might you draw from your evaluations?


5f

217. Pulleys A block-and-tackle system is designed so that pulling 5 feet on one end of a rope will move a weight on the other end a distance of 3 feet. If x represents the distance the rope is pulled, express the distance the weight moves in terms of x.

t

On the basis of your answers, make a conjecture that appears to be true about the expressions xn and nx, where n 3, 4, 5, 6, 7, . . . and x is a natural number greater than 2.

Section 1.5 / Sets

1.5 Objective A

55

Sets To write a set using the roster method

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A set is a collection of objects, which are called the elements of the set. The roster method of writing a set encloses a list of the elements in braces. The set of the last three letters of the alphabet is written {x, y, z}. The set of the positive integers less than 5 is written {1, 2, 3, 4}. HOW TO

Use the roster method to write the set of integers between 0 and 10.

A {1, 2, 3, 4, 5, 6, 7, 8, 9} A set can be designated by a capital letter. Note that 0 and 10 are not elements of set A. HOW TO

Use the roster method to write the set of natural numbers.

A {1, 2, 3, 4, . . .}

• The three dots mean that the pattern of numbers continues without end.

The empty set, or null set, is the set that contains no elements. The symbol or { } is used to represent the empty set. The set of people who have run a two-minute mile is the empty set. The union of two sets, written A B, is the set that contains the elements of A and the elements of B. HOW TO

Find A B, given A {1, 2, 3, 4} and B {3, 4, 5, 6}.

A B {1, 2, 3, 4, 5, 6}

• The union of A and B contains all the elements of A and all the elements of B. Any elements that are in both A and B are listed only once.


The intersection of two sets, written A B, is the set that contains the elements that are common to both A and B. HOW TO


A B {3, 4}

• The intersection of A and B contains the elements common to A and B.

Example 1

You Try It 1

Use the roster method to write the set of the odd positive integers less than 12.

Use the roster method to write the set of the odd negative integers greater than 10.

Solution

Your solution

A {1, 3, 5, 7, 9, 11} Solution on p. S3

56


Example 2

You Try It 2

Use the roster method to write the set of the even positive integers.

Use the roster method to write the set of the odd positive integers.

Solution

Your solution

A {2, 4, 6, . . .}

Example 3

You Try It 3

Find D E, given D {6, 8, 10, 12} and E {8, 6, 10, 12}.

Find A B, given A {2, 1, 0, 1, 2} and B {0, 1, 2, 3, 4}.

Solution

Your solution

D E {8, 6, 6, 8, 10, 12}

Example 4

You Try It 4


Find C D, given C {10, 12, 14, 16} and D {10, 16, 20, 26}.

Solution

Your solution

A B {5, 9}

Example 5

You Try It 5



Solution

Your solution

AB

Objective B

Point of Interest The symbol was first used in the book Arithmeticae Principia, published in 1889. It was the first letter of the Greek word , which means “is.” The symbols for union and intersection were also introduced at that time.

To write a set using set-builder notation

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SSM

Another method of representing sets is called set-builder notation. Using setbuilder notation, the set of all positive integers less than 10 is written as follows: {x x 10, x positive integers}, which is read “the set of all x such that x is less than 10 and x is an element of the positive integers.”

HOW TO than 4.

Use set-builder notation to write the set of real numbers greater

{x x 4, x real numbers}

• “x real numbers” is read “x is an element of the real numbers.”


Solutions on p. S3

57

Section 1.5 / Sets

Example 6

You Try It 6

Use set-builder notation to write the set of negative integers greater than 100.

Use set-builder notation to write the set of positive even integers less than 59.

Solution

Your solution

{xx 100, x negative integers} Example 7

You Try It 7

Use set-builder notation to write the set of real numbers less than 60.

Use set-builder notation to write the set of real numbers greater than 3.

Solution

Your solution

{xx 60, x real numbers} Solutions on p. S3

Objective C

Point of Interest The symbols for “is less than” and “is greater than” were introduced by Thomas Harriot around 1630. Before that, and were used for and , respectively.

TA K E N O T E


In many cases, we assume that the real numbers are being used and omit “x real numbers” from set-builder notation. Using this convention, {x x 1, x real numbers} is written {x x 1}.

To graph an inequality on the number line An expression that contains the symbol , , , or is called an inequality. An inequality expresses the relative order of two mathematical expressions. The expressions can be either numerical or variable.

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4 2

   Inequalities 3x 7   2 x 2x y 4 

An inequality can be graphed on the number line. HOW TO

Graph: {x x 1}

The graph is the real numbers greater than 1. The parenthesis at 1 indicates that 1 is not included in the graph. HOW TO

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

Graph: {x x 1}

The bracket at 1 indicates that 1 is included in the graph. HOW TO

−5 −4 −3 −2 −1

Graph: {x x 1} −5 −4 −3 −2 −1

0

1

2

3

4

5

4

5

The numbers less than 1 are to the left of 1 on the number line. The union of two sets is the set that contains all the elements of each set. HOW TO

Graph: {x x 4} {x x 1}

The graph is the numbers greater than 4 and the numbers less than 1.

−5 −4 −3 −2 −1

0

1

2

3

58


The intersection of two sets is the set that contains the elements common to both sets. HOW TO

Graph: {x x 1} {x x 2}

The graphs of {x x 1} and {x x 2} are shown at the right.

The graph of {x x 1} {x x 2} is the numbers between 1 and 2.

–5 –4 –3 –2 –1

0

1

2

3

4

5

–5 –4 –3 –2 –1

0

1

2

3

4

5

–5 –4 –3 –2 –1

0

1

2

3

4

5

Example 8

You Try It 8

Graph: {x x 5}

Graph: {x x 2}

Solution

Your solution

The graph is the numbers less than 5. −5 − 4 − 3 − 2 −1

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

Example 9

You Try It 9

Graph: { xx 3} {xx 1}

Graph: {xx 1} {xx 3}

Solution

Your solution

The graph is the numbers greater than 3 and the numbers less than 1. −5 − 4 − 3 − 2 − 1

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

Example 10

You Try It 10



Solution

Your solution

−5 −4 −3 −2 −1

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

Example 11

You Try It 11



Solution

Your solution

The graph is the real numbers. −5 − 4 − 3 − 2 − 1

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

Solutions on p. S3


The graph is the numbers between 2 and 1.

Section 1.5 / Sets

59


To write a set using the roster method

Use the roster method to write the set. 01. the integers between 15 and 22

02. the integers between 10 and 4

03. the odd integers between 8 and 18

04. the even integers between 11 and 1

05. the letters of the alphabet between a and d

06. the letters of the alphabet between p and v

07.

Explain how to find the union of two sets.

08.

Explain how to find the intersection of two sets.

Find A B. 09. A {3, 4, 5}

B {4, 5, 6}

10. A {3, 2, 1}

B {2, 1, 0}

11. A {10, 9, 8}

B {8, 9, 10}

12. A {a, b, c}

B {x, y, z}

13. A {a, b, d, e}

B {c, d, e, f}

14. A {m, n, p, q}

B {m, n, o}

15. A {1, 3, 7, 9}

B {7, 9, 11, 13}

16. A {3, 2, 1}

B {1, 1, 2}

17. A {3, 4, 5}

B {4, 5, 6}

18. A {4, 3, 2}

B {6, 5, 4}

19. A {4, 3, 2}

B {2, 3, 4}

20. A {1, 2, 3, 4}

B {1, 2, 3, 4}

21. A {a, b, c, d, e}

B {c, d, e, f, g}

22. A {m, n, o, p}

B {k, l, m, n}


Find A B.

Objective B

To write a set using set-builder notation

Use set-builder notation to write the set. 23. the negative integers greater than 5

24. the positive integers less than 5

60


25. the integers greater than 30

26. the integers less than 70

27. the even integers greater than 5

28. the odd integers less than 2

29. the real numbers greater than 8

30. the real numbers less than 57

Objective C

To graph an inequality on the number line

Graph. 31. {xx 2} −5 − 4 − 3 − 2 −1

32. 0

1

2

3

4

− 5 −4 −3 − 2 − 1

34. 0

1

2

3

4

0

1

2

3

4

36.

37. {xx 2} {xx 4} −5 −4 −3 −2 −1

0

1

2

3

38. 4

5

39. {xx 2} {xx 4} −5 −4 − 3 − 2 − 1

0

1

2

3

4

5

2

3

4

5

0

1

2

3

4

5

4

5

4

5

4

5

0

1

2

3

{xx 3} {xx 3} −5 −4 −3 −2 −1

40.

1

{xx 4} {xx 2} −5 −4 −3 −2 −1

5

0

{xx 4} −5 −4 −3 −2 −1

5

35. {xx 2} {xx 4} −5 − 4 − 3 − 2 − 1

−5 −4 −3 −2 −1

5

33. {xx 0}

{xx 1}

0

1

2

3

{xx 0} {xx 4} −5 −4 − 3 −2 −1

0

1

2

3

41. Determine whether the statement is always true, sometimes true, or never true. a. Given that a 0 and b 0, then ab 0. b. Given that a 0, then a2 0. c. Given that a 0 and b 0, then a2 b. 42. By trying various sets, make a conjecture as to whether the union of two sets is a. a commutative operation b. an associative operation 43. By trying various sets, make a conjecture as to whether the intersection of two sets is a. a commutative operation b. an associative operation




61

Focus on Problem Solving Inductive Reasoning

Suppose you take 9 credit hours each semester. The total number of credit hours you have taken at the end of each semester can be described in a list of numbers. 9, 18, 27, 36, 45, 54, 63, . . . The list of numbers that indicates the total credit hours is an ordered list of numbers, called a sequence. Each number in a sequence is called a term of the sequence. The list is ordered because the position of a number in the list indicates after which semester that number of credit hours has been taken. For example, the 7th term of the sequence is 63, and a total of 63 credit hours have been taken after the 7th semester. Assuming the pattern continues, find the next three numbers in the pattern 6, 10, 14, 18, . . . This list of numbers is a sequence. The first step in solving this problem is to observe the pattern in the list of numbers. In this case, each number in the list is 4 less than the previous number. The next three numbers are 22, 26, 30. This process of discovering the pattern in a list of numbers uses inductive reasoning. Inductive reasoning involves making generalizations from specific examples; in other words, we reach a conclusion by making observations about particular facts or cases. Try the following exercises. Each exercise requires inductive reasoning. Name the next two terms in the sequence. 1. 1, 3, 5, 7, 1, 3, 5, 7, 1, . . .

2. 1, 4, 2, 5, 3, 6, 4, . . .

3. 1, 2, 4, 7, 11, 16, . . .

4. A, B, C, G, H, I, M, . . .

Draw the next shape in the sequence.


5.

6. Solve. 7. Convert convert

8. Convert convert

1 2 3 4 5 , , , , and to decimals. 11 11 11 11 11 6 7 9 , , and to decimals. 11 11 11 1 2 4 5 7 , , , , and to decimals. 33 33 33 33 33 19 8 13 , , and to decimals. 33 33 33

Then use the pattern you observe to

Then use the pattern you observe to


Projects and Group Activities Calculators

Does your calculator use the Order of Operations Agreement? To find out, try this problem: 247 If your answer is 30, then the calculator uses the Order of Operations Agreement. If your answer is 42, it does not use the agreement. Even if your calculator does not use the Order of Operations Agreement, you can still correctly evaluate numerical expressions. The parentheses keys, ( and ) , are used for this purpose. Remember that 2 4 7 means 2 (4 7) because the multiplication must be completed before the addition. To evaluate this expression, enter the following: Enter:

2

(

4

7

)

Display:

2

2

(

4

4

7

28

30

When using your calculator to evaluate numerical expressions, insert parentheses around multiplications or divisions. This has the effect of forcing the calculator to do the operations in the order you want rather than in the order the calculator wants. Evaluate. 1. 3 (15 2 3) 36 3 3. 16 4 3 (3 4 5) 2

2. 4 22 (12 24 6) 5 4. 15 3 9 (2 6 3) 4

Using your calculator to simplify numerical expressions sometimes requires use of the key or, on some calculators, the negative key, which is frequently shown as () . These keys change the sign of the number currently in the display. To enter 4: • For those calculators with , press 4 and then . • For those calculators with () , press () and then 4. Here are the keystrokes for evaluating the expression 3(4) (5). Calculators with key:

3 4 5

Calculators with () key:

3

() 4

() 5

This example illustrates that calculators make a distinction between negative and minus. To perform the operation 3 (3), you cannot enter 3 3. This would result in 0, which is not the correct answer. You must enter 3 3

or

3

() 3

Use a calculator to evaluate each of the following exercises. 5. 16 2 8. 50 (14)

6. 3(8) 9. 4 (3)2

7. 47 (9) 10. 8 (6)2 7


62

Chapter 1 Summary

63


Examples

The set of natural numbers is 1, 2, 3, 4, 5, . . .. The set of integers is . . . , 3, 2, 1, 0, 1, 2, 3, . . .. [1.1A, p. 3] A number a is less than a number b, written a b, if a is to the left of b on a number line. A number a is greater than a number b, written a b, if a is to the right of b on a number line. The symbol means is less than or equal to. The symbol means is greater than or equal to. [1.1A, p. 3]

5 3 3 3 55

Two numbers that are the same distance from zero on the number line but on opposite sides of zero are opposite numbers or opposites. The additive inverse of a number is the opposite of the number. [1.1B, p. 4]

7 and 7 are opposites.

The absolute value of a number is its distance from 0 on the number line. [1.1B, p. 4]

5 5

A rational number (or fraction) is a number that can be written

3 , 8

in the form

a , b

where a and b are integers and b 0. A rational


number can be represented as a terminating or repeating decimal. [1.2A, p. 17]

3 4

and

3 4

9 0 4 7 6 9

are opposites.

2.3 2.3

0 0

9 2

, and 4 are rational numbers.

1.13 and 0.473 are also rational numbers.

Percent means “parts of 100.” [1.2B, p. 18]

72% means 72 of 100 equal parts.

An expression of the form an is in exponential form. The base is a and the exponent is n. [1.2E, p. 22]

54 is an exponential expression. The base is 5 and the exponent is 4.

A square root of a positive number x is a number whose square is x. The principal square root of a number is the positive square root. The symbol is called a radical sign and is used to indicate the principal square root of a number. The radicand is the number under the radical sign. [1.2F, p. 24]

25 5 25 5

The square of an integer is a perfect square. If a number is not a perfect square, its square root can only be approximated. [1.2F, p. 24]

72 49; 49 is a perfect square.

An irrational number is a number that has a decimal representation that never terminates or repeats. [1.2F, p. 24]

, 2, and 1.34334333433334 . . . are irrational numbers.

The rational numbers and the irrational numbers taken together are the real numbers. [1.2F, p. 24]

3 , 8

9 2

, 4, 1.13, 0.473, , 2, and

1.34334333433334 . . . are real numbers.

64


A variable is a letter that is used for a quantity that is unknown or that can change. A variable expression is an expression that contains one or more variables. [1.4A, p. 37]

4x 2y 6z is a variable expression. It contains the variables x, y, and z.

The terms of a variable expression are the addends of the expression. Each term is a variable term or a constant term. [1.4A, p. 37]

The expression 2a2 3b3 7 has three terms, 2a2, 3b3, and 7. 2a2 and 3b3 are variable terms. 7 is a constant term.

A variable term is composed of a numerical coefficient and a variable part. [1.4A, p. 37]

For the expression 7x3y2, 7 is the coefficient and x3y2 is the variable part.

In a variable expression, replacing each variable by its value and then simplifying the resulting numerical expression is called evaluating the variable expression. [1.4A, p. 37]

To evaluate 2ab b2 when a 3 and b 2, replace a by 3 and b by 2 and then simplify the numerical expression. 232 22 16

Like terms of a variable expression are terms with the same variable part. Constant terms are like terms. [1.4B, p. 39]

For the expressions 3a2 2b 3 and 2a2 3a 4, 3a2 and 2a2 are like terms; 3 and 4 are like terms.

To simplify the sum of like variable terms, use the Distributive Property to add the numerical coefficients. This is called combining like terms. [1.4B, p. 39]

5y 3y 5 3y 8y

The multiplicative inverse of a number is the reciprocal of the number. [1.4C, p. 41]

3 4

4 3

is the multiplicative inverse of . 1 4

A set is a collection of objects, which are called the elements of the set. The roster method of writing a set encloses a list of the elements in braces. The empty set or null set, written , is the set that contains no elements. [1.5A, p. 55]

The set of cars that can travel faster than 1000 mph is an empty set.

The union of two sets, written A B, is the set that contains the elements of A and the elements of B. [1.5A, p. 55]

Let A 2, 4, 6, 8 and B 0, 1, 2, 3, 4. Then A B 0, 1, 2, 3, 4, 6, 8.

The intersection of two sets, written A B, is the set that contains the elements that are common to both A and B. [1.5A, p. 55]

Let A 2, 4, 6, 8 and B 0, 1, 2, 3, 4. Then A B 2, 4.

Set-builder notation uses a rule to describe the elements of a set. [1.5B, p. 56]

Using set-builder notation, the set of real numbers greater than 2 is written xx 2, x real numbers.

Essential Rules and Procedures

Examples

To add two numbers with the same sign, add the absolute

7 15 22 7 15 22

values of the numbers. Then attach the sign of the addends. [1.1C, p. 5]


is the multiplicative inverse of 4.

Chapter 1 Summary

To add two numbers with different signs, find the absolute

value of each number. Subtract the smaller of the two numbers from the larger. Then attach the sign of the number with the larger absolute value. [1.1C, p. 5]

7 15 8 7 15 8

To subtract one number from another, add the opposite of the second number to the first number. [1.1C, p. 6]

7 19 7 19 12 6 13 6 13 7

To multiply two numbers with the same sign, multiply the absolute values of the numbers. The product is positive. [1.1D, p. 8]

7 8 56 78 56

To multiply two numbers with different signs, multiply the absolute values of the numbers. The product is negative. [1.1D, p. 8]

7 8 56 78 56

To divide two numbers with the same sign, divide the absolute values of the numbers. The quotient is positive. [1.1D, p. 9]

54 9 6 54 9 6

To divide two numbers with different signs, divide the absolute values of the numbers. The quotient is negative. [1.1D, p. 9]

54 9 6 54 9 6

Properties of Zero and One in Division [1.1D, p. 9]

0 0 5 12 1 12 7 7 1

0 0. a a 0, 1. a

If a 0, If a a 1 a 0

a is undefined.

8 0

To write a percent as a fraction, remove the percent sign and


multiply by

1 . 100

[1.2B, p. 18]

To write a percent as a decimal, remove the percent sign and

multiply by 0.01. [1.2B, p. 18] To write a decimal or a fraction as a percent, multiply by

100%. [1.2B, p. 18]

is undefined.

60% 60

1 100

60 3 100 5

73% 730.01 0.73 1.3% 1.30.01 0.013 0.3 0.3100% 30% 5 500 5 100% % 62.5% 8 8 8

To add two fractions with the same denominator, add the numerators and place the sum over the common denominator. [1.2C, p. 19]

7 1 71 8 4 10 10 10 10 5

To subtract two fractions with the same denominator,

7 1 71 6 3 10 10 10 10 5

subtract the numerators and place the difference over the common denominator. [1.2C, p. 19]

65

66


To multiply two fractions, place the product of the numerators over the product of the denominators. [1.2D, p. 20]

25 10 5 2 5 3 6 36 18 9

To divide two fractions, multiply the dividend by the

4 2 4 2 223 6 5 3 5 3 52 5

reciprocal of the divisor. [1.2D, p. 21] Product Property of Square Roots [1.2F, p. 25]

ab a b

50 25 2 25 2 5 2

Order of Operations Agreement [1.3A, p. 33] Step 1 Perform operations inside grouping symbols. Grouping

symbols include parentheses ( ), brackets [ ], braces { }, and the fraction bar. Step 2 Simplify exponential expressions.

50 52 27 16 50 52 29 50 25 29 2 18 16

Step 3 Do multiplication and division as they occur from left

to right. Step 4 Do addition and subtraction as they occur from left

to right.

If a, b, and c are real numbers, then ab c ab ac. The Associative Property of Addition [1.4B, p. 39]

If a, b, and c are real numbers, then a b c a b c.

54 7 5 4 5 7 20 35 55 4 2 7 4 9 5 4 2 7 2 7 5

The Commutative Property of Addition [1.4B, p. 39] If a and b are real numbers, then a b b a.

2 5 7 and 5 2 7

The Addition Property of Zero [1.4B, p. 39] If a is a real number, then a 0 0 a a.

8 0 8 and 0 8 8

The Inverse Property of Addition [1.4B, p. 40] If a is a real number, then a a a a 0.

5 5 0 and

The Associative Property of Multiplication [1.4C, p. 40]

If a, b, and c are real numbers, then abc abc.

3 5 4 320 60 3 5 4 15 4 60

The Commutative Property of Multiplication [1.4C, p. 41]

37 21 and 73 21

If a and b are real numbers, then ab ba.

5 5 0

The Multiplication Property of One [1.4C, p. 41] If a is a real number, then a 1 1 a a.

31 3 and 13 3

The Inverse Property of Multiplication [1.4C, p. 41]

3

If a is a real number and a is not equal to zero, then a

1 a

1 a

a 1.

1 1 and 3

1 3 1 3


The Distributive Property [1.4B, p. 38]

Chapter 1 Review Exercises

Chapter 1 Review Exercises 1. Let x {4, 0, 11}. For what values of x is the inequality x 1 a true statement?

2. Find the additive inverse of 4.

3. Evaluate 5.

4. Add: 3 (12) 6 (4)

5. Subtract: 16 (3) 18

6. Multiply: (6)(7)

7. Divide: 100 5

8. Write


9. Write 6.2% as a decimal.

10. Write

7 25

5 8

as a decimal.

as a percent.

11. Simplify:

1 1 5 3 6 12

12. Subtract: 5.17 6.238

13. Divide:

17 18 35 28

14. Multiply: 4.32(1.07)

16. Simplify: 2 36

15. Evaluate

2 3

4

.

17. Simplify: 3 120

18. Evaluate 32 4[18 (12 20)].

67

68


19. Evaluate (b a)2 c when a 2, b 3, and c 4.

20.

Simplify: 6a 4b 2a

21. Simplify: 3(12y)

22.

Simplify: 5(2x 7)

23. Simplify: 4(2x 9) 5(3x 2)

24.

Simplify: 5[2 3(6x 1)]

25. Use the roster method to write the set of odd positive integers less than 8.

26.

Find A B, given A [1, 5, 9, 13} and B {1, 3, 5, 7, 9}.

27. Graph {xx 3}.

28.

Graph {xx 3} {xx 2}.

−5 − 4 − 3 − 2 −1

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

29. Testing To discourage random guessing on a multiple-choice exam, a professor assigns 6 points for a correct answer, 4 points for an incorrect answer, and 2 points for leaving a question blank. What is the score for a student who had 21 correct answers, had 5 incorrect answers, and left 4 questions blank?

30. Candy The circle graph shows the amount of candy consumed by Americans during a recent year (Source: Candy USA). What percent of the candy consumed was chocolate? Round to the nearest tenth of a percent.

0.5 billion pounds Gum

Nonchocolate 2.7 billion pounds

Chocolate 3.3 billion pounds

32. Baseball Cards A baseball card collection contains five times as many National League players’ cards as American League players’ cards. Express the number of National League players’ cards in terms of the number of American League players’ cards.

33. Money A club treasurer has some five-dollar bills and some ten-dollar bills. The treasurer has a total of 35 bills. Express the number of fivedollar bills in terms of the number of ten-dollar bills.


31. Translate “the difference between twice a number and one-half of the number” into a variable expression. Then simplify.

Chapter 1 Test

69

Chapter 1 Test 1. Place the correct symbol, or , between the two numbers. 2 40

2.

Find the opposite of 4.

3. Evaluate 4.

4.

Subtract: 16 30

5. Add: 22 14 (8)

6.

Subtract: 16 (30) 42

7. Divide: 561 (33)

8.

Write

7 9


repeating digit of the decimal.

10.

7 2 Add: 5 15

12.

Divide:

14.

Simplify: 2 45

15. Evaluate 16 2[8 3(4 2)] 1.

16.

Evaluate b2 3ab when a 3 and b 2.

17. Simplify: 3x 5x 7x

18.

Simplify:

9. Write 45% as a fraction and as a decimal.


11. Multiply: 6.02(0.89)

13. Evaluate

3 (4)2. 4

5 5 12 6

1 (10x) 5

70


19. Simplify: 3(2x2 7y 2)

20.

Simplify: 2x 3(x 2)

21. Simplify: 2x 3[4 (3x 7)]

22.

Use the roster method to write the set of integers between 3 and 4.

23. Use set-builder notation to write the set of real numbers less than 3.

24.

Find A B given A {1, 3, 5, 7} and B {2, 4, 6, 8}.

25. Graph {x x 1}.

26.

Graph {x x 3} {xx 0}.

−5 − 4 − 3 − 2 −1

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

28. Baseball The speed of a pitcher’s fastball is twice the speed of the catcher’s return throw. Express the speed of the fastball in terms of the speed of the return throw.

29.

Balance of Trade The table at the right shows the U.S. balance of trade, in billions of dollars, for the years 1980 to 2000 (Source: U.S. Dept. of Commerce). a. In which years did the trade balance increase from the previous year? b. Calculate the difference between the trade balance in 1990 and the trade balance in 2000. c. During which two consecutive years was the difference in the trade balance greatest? d. How many times greater was the trade balance in 1990 than in 1980? Round to the nearest whole number. e. Calculate the average trade balance per quarter for the year 2000.

30.

Temperature The lowest temperature recorded in North America is 81.4°F. The highest temperature recorded is 134.0°F (Source: National Climatic Data Center). Find the difference beween these two extremes.

Year

Trade Balance

1980

−19.4

1981

−16.2

1982

−24.2

1983

−57.8

1984

−109.2

1985

−122.1

1986

−140.6

1987

−153.3

1988

−115.9

1989

−92.2

1990

−81.1

1991

−30.7

1992

−35.7

1993

−68.9

1994

−97.0

1995

−95.9

1996

−102.1

1997

−104.7

1998

−166.9

1999

−265.0

2000

−369.7


27. Translate “ten times the difference between a number and 3” into a variable expression. Then simplify.

chapter

2

First-Degree Equations and Inequalities

OBJECTIVES

Section 2.1

A B C D E

To determine whether a given number is a solution of an equation To solve an equation of the form xab To solve an equation of the form ax b To solve application problems using the basic percent equation To solve uniform motion problems

Section 2.2

A


B Hourly wage, salary, and commissions are three ways to receive payment for doing work. Commissions are usually paid to salespersons and are calculated as a percent of total sales. The salesperson in this photo receives a combination of an hourly wage and commissions. The sales personnel in Exercises 94 and 95 on page 135 receive a combination of salary and commissions. In these exercises, you will be using first-degree inequalities to determine the amount of sales needed to reach target income goals.

C D

To solve an equation of the form ax b c To solve an equation of the form ax b cx d To solve an equation containing parentheses To solve application problems using formulas

Section 2.3

A B

To solve integer problems To translate a sentence into an equation and solve

Section 2.4

A B C

To solve value mixture problems To solve percent mixture problems To solve uniform motion problems

Section 2.5

A B C

To solve an inequality in one variable To solve a compound inequality To solve application problems

Section 2.6 Need help? For online student resources, such as section quizzes, visit this textbook’s website at math.college.hmco.com/students.

A B C

To solve an absolute value equation To solve an absolute value inequality To solve application problems

PREP TEST Do these exercises to prepare for Chapter 2. Write

as a decimal.

3.

Evaluate 3x2 4x 1 when x 4.

5.

Simplify:

7.

Simplify: 0.223x 6 x

9.

A new graphics card for computer games is five times faster than a graphics card made two years ago. Express the speed of the new card in terms of the speed of the old card.

1 2 x x 2 3

3 4

2.

Write

as a percent.

4.

Simplify: R 0.35R

6.

Simplify: 6x 36 x

8.

Translate into a variable expression: “The difference between 5 and twice a number.”

10. A board 5 ft long is cut into two pieces. If x represents the length of the longer piece, write an expression for the shorter piece in terms of x.

GO FIGURE How can a donut be cut into eight equal pieces with three cuts of a knife?


9 100

1.

Section 2.1 / Introduction to Equations

2.1 Objective A

Point of Interest One of the most famous equations ever stated is E mc 2. This equation, stated by Albert Einstein, shows that there is a relationship between mass m and energy E. As a side note, the chemical element einsteinium was named in honor of Einstein.

73

Introduction to Equations To determine whether a given number is a solution of an equation

VIDEO & DVD

CD TUTOR

SSM

WEB

An equation expresses the equality of two mathematical expressions. The expressions can be either numerical or variable expressions.

 9 3 12   3x 2 10  Equations y2 4 2y 1   z2 

The equation at the right is true if the variable is replaced by 5.

x 8 13 5 8 13

A true equation

The equation is false if the variable is replaced by 7.

7 8 13

A false equation

A solution of an equation is a number that, when substituted for the variable, results in a true equation. 5 is a solution of the equation x 8 13. 7 is not a solution of the equation x 8 13. HOW TO TA K E N O T E The Order of Operations Agreement applies to evaluating 22 5 and 22 3.

Example 1


Solution

Is 2 a solution of 2x 5 x2 3?

2x 5 x2 3 22 5 22 3 4 5 4 3 11 Yes, 2 is a solution of the equation.

• Replace x by 2. • Evaluate the numerical expressions. • If the results are equal, 2 is a solution of the equation. If the results are not equal, 2 is not a solution of the equation.


You Try It 1

5x 2 6x 2 54 2 64 2 20 2 24 2 22 22

Your solution

Is

1 4

a solution of

5 4x 8x 2?

Yes, 4 is a solution. Example 2

Is 4 a solution of 4 5x x2 2x?

You Try It 2

Solution

4 5x x2 2x 4 54 42 24 4 20 16 8 16 24

Your solution

Is 5 a solution of 10x x2 3x 10?

( means “is not equal to”) No, 4 is not a solution. Solutions on p. S3


Objective B

Study

Tip

To learn mathematics, you must be an active participant. Listening and watching your professor do mathematics is not enough. Take notes in class, mentally think through every question your instructor asks, and try to answer it even if you are not called on to answer it verbally. Ask questions when you have them. See AIM for Success, page xxvii, for other ways to be an active learner.

To solve an equation of the form x a b

VIDEO & DVD

CD TUTOR

SSM

WEB

To solve an equation means to find a solution of the equation. The simplest equation to solve is an equation of the form variable constant, because the constant is the solution. The solution of the equation x 5 is 5 because 5 5 is a true equation. The solution of the equation at the right is 7 because 7 2 9 is a true equation.

x29

729

Note that if 4 is added to each side of the equation x 2 9, the solution is still 7.

x29 x2494 x 6 13

7 6 13

If 5 is added to each side of the equation x 2 9, the solution is still 7.

x29 x 2 5 9 5 x34

7 3 43

Equations that have the same solution are equivalent equations. The equations x 2 9, x 6 13, and x 3 4 are equivalent equations; each equation has 7 as its solution. These examples suggest that adding the same number to each side of an equation produces an equivalent equation. This is called the Addition Property of Equations.

Addition Property of Equations

The same number can be added to each side of an equation without changing its solution. In symbols, the equation a b has the same solution as the equation a c b c.

In solving an equation, the goal is to rewrite the given equation in the form variable constant. The Addition Property of Equations is used to remove a term from one side of the equation by adding the opposite of that term to each side of the equation. HOW TO

Solve: x 4 2

x42 x4424 x06 x6

• The goal is to rewrite the equation as variable constant . • Add 4 to each side of the equation. • Simplify. • The equation is in the form variable constant .

Check: x 4 2 642 22

A true equation

The solution is 6. Because subtraction is defined in terms of addition, the Addition Property of Equations also makes it possible to subtract the same number from each side of an equation without changing the solution of the equation.


74


HOW TO

Solve: y

3 1 4 2 3 3 1 3 y 4 4 2 4 3 2 y0 4 4 1 y 4 y

75

3 1 4 2 • The goal is to rewrite the equation in the form variable constant . 3 4

• Subtract from each side of the equation. • Simplify. • The equation is in the form variable constant .

1 4

The solution is . You should check this solution.

Example 3

Solve: x

2 1 5 3

You Try It 3

Solution

2 5

• Subtract from each side.

1 . 15

Objective C

Solution on p. S3

To solve an equation of the form ax b The solution of the equation at the right is 3 because 2 3 6 is a true equation.


5 3 y 6 8

Your solution

2 1 x 5 3 2 1 2 2 x 5 5 3 5 5 6 x0 15 15 1 x 15 The solution is

Solve:

Note that if each side of 2x 6 is multiplied by 5, the solution is still 3. If each side of 2x 6 is multiplied by 4, the solution is still 3.

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2x 6

236

2x 6 52x 5 6 10x 30

10 3 30

2x 6 42x 46 8x 24

8 3 24

The equations 2x 6, 10x 30, and 8x 24 are equivalent equations; each equation has 3 as its solution. These examples suggest that multiplying each side of an equation by the same nonzero number produces an equivalent equation. Multiplication Property of Equations

Each side of an equation can be multiplied by the same nonzero number without changing the solution of the equation. In symbols, if c 0, then the equation a b has the same solutions as the equation ac bc.

76


The Multiplication Property of Equations is used to remove a coefficient by multiplying each side of the equation by the reciprocal of the coefficient. HOW TO

Solve:

3 z9 4 4 3 4 z 9 3 4 3 1 z 12 z 12

3 z9 4 • The goal is to rewrite the equation in the form variable constant . 4 3

• Multiply each side of the equation by . • Simplify. • The equation is in the form variable constant .

The solution is 12. You should check this solution. Because division is defined in terms of multiplication, each side of an equation can be divided by the same nonzero number without changing the solution of the equation. TA K E N O T E Remember to check the solution. Check:

6x 14 7 14 6 3 14 14

HOW TO

Solve: 6x 14

6x 14 6x 14 6 6 7 x 3

• The goal is to rewrite the equation in the form variable constant . • Divide each side of the equation by 6. • Simplify. The equation is in the form variable constant . 7 3

The solution is . When using the Multiplication Property of Equations, multiply each side of the equation by the reciprocal of the coefficient when the coefficient is a fraction. Divide each side of the equation by the coefficient when the coefficient is an integer or a decimal.

Solution

Solve:

3x 9 4

You Try It 4

3x 9 4 4 4 3 x 9 3 4 3 x 12

Solve:

2x 6 5

Your solution •

3x 3 x 4 4

The solution is 12. Example 5

Solve: 5x 9x 12

You Try It 5

Solution

5x 9x 12 4x 12 4x 12 4 4 x 3

Your solution

Solve: 4x 8x 16


The solution is 3.

Solutions on pp. S3–S4


Example 4


Objective D

To solve application problems using the basic percent equation

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An equation that is used frequently in mathematics applications is the basic percent equation.

Basic Percent Equation

Percent Base Amount P

B

A

In many application problems involving percent, the base follows the word of. HOW TO

20% of what number is 30?

PBA 0.20B 30 0.20B 30 0.20 0.20 B 150

• Use the basic percent equation. • P 20% 0.20, A 30, and B is unknown. • Solve for B.

The number is 150.

TA K E N O T E We have written P80 70 because that is the form of the basic percent equation. We could have written 80P 70. The important point is that each side of the equation is divided by 80, the coefficient of P.

HOW TO

70 is what percent of 80?

PBA P80 70 P80 70 80 80 P 0.875 P 87.5%

• Use the basic percent equation. • B 80, A 70, and P is unknown. • Solve for P. • The question asked for a percent. Convert the decimal to a percent.


70 is 87.5% of 80.

HOW TO The world’s production of cocoa for a recent year was 2928 metric tons. Of this, 1969 metric tons came from Africa. (Source: World Cocoa Foundation) What percent of the world’s cocoa production came from Africa? Round to the nearest tenth of a percent. Strategy

To find the percent, use the basic percent equation. B 2928, A 1969, P is unknown.

Solution

PBA P2928 1969 P

1969 0.672 2928

Approximately 67.2% of the world’s cocoa production came from Africa.

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The simple interest that an investment earns is given by the simple interest equation I Prt, where I is the simple interest, P is the principal, or amount invested, r is the simple interest rate, and t is the time.

A $1500 investment has an annual simple interest rate of 7%. HOW TO Find the simple interest earned on the investment after 18 months. The time is given in months but the interest rate is an annual rate. Therefore, we must convert 18 months to years. 18 months

18 years 1.5 years 12

To find the interest, solve I Prt for I. I Prt I 15000.071.5 I 157.5

• P 1500, r 0.07, t 1.5

The investment earned $157.50.

In the jewelry industry, the amount of gold in a piece of jewelry is measured by the karat. Pure gold is 24 karats. A necklace that is 18 karats is 18 0.75 75% gold. 24

The amount of a substance in a solution can be given as a percent of the total solution. For instance, if a certain fruit juice drink is advertised as containing 27% cranberry juice, then 27% of the contents of the bottle must be cranberry juice.

When solving problems involving mixtures, we use the percent mixture equation Q Ar, where Q is the quantity of a substance in the solution, A is the amount of the solution, and r is the percent concentration of the substance.

Part of the formula for a perfume requires that the concentration HOW TO of jasmine be 1.2% of the total amount of perfume. How many ounces of jasmine are in a 2-ounce bottle of this perfume? The amount of perfume is 2 oz. Therefore, A 2. The percent concentration is 1.2%, so r 0.012. To find the number of ounces of jasmine, solve Q Ar for Q. Q Ar Q 20.012 Q 0.024

• A 2, r 0.012

There is 0.024 ounce of jasmine in the perfume.

In most cases, you should write the percent as a decimal before solving the basic percent equation. However, some percents are more easily written as fractions. For example, 1 1 33 % 3 3

2 2 66 % 3 3

2 1 16 % 3 6

1 5 83 % 3 6


Point of Interest


Example 6

You Try It 6

1 3

2 3

12 is 33 % of what number?

18 is 16 % of what number?

Solution

Your solution

PBA 1 B 12 3 1 3 B 3 12 3 B 36

79

• Use the basic percent equation. 1 3

• 33 %

1 3

1 3

12 is 33 % of 36.

Example 7

You Try It 7

The data in the table below show the number of households (in millions) that downloaded music files for a three-month period in a recent year. (Source: NPD Group)

The Bowl Championship Series (BCS) received approximately $83.3 million in revenues from various college football bowl games. Of this amount, the college representing the Pac-10 in the Rose Bowl received approximately $3.1 million. (Source: BCSfootball.org) What percent of the total received by the BCS did the college representing the Pac-10 receive? Round to the nearest tenth of a percent.

Month

April

May

June

Downloads

14.5

12.7

10.4

For the three-month period, what percent of the files were downloaded in May? Round to the nearest percent. Strategy

Your strategy


To find the percent, Find the total number of files downloaded for the three-month period. Use the basic percent equation. B is the total number of files downloaded for the threemonth period; A 12.7, the number of files downloaded in May; P is unknown. Your solution

Solution

14.5 12.7 10.4 37.6 PBA P37.6 12.7 12.7 P 0.34 37.6

• Use the basic percent equation. • B 37.6, A 12.7

Approximately 34% of the files were downloaded in May. Solutions on p. S4

80


Example 8

You Try It 8

In April, Marshall Wardell was charged an interest fee of $8.72 on an unpaid credit card balance of $545. Find the annual interest rate on this credit card.

Clarissa Adams purchased a municipal bond for $1000 that earns an annual simple interest rate of 6.4%. How much must she deposit into a bank account that earns 8% annual simple interest so that the interest earned from each account after one year is the same?

Strategy

Your strategy

The interest is $8.72. Therefore, I 8.72. The unpaid balance is $545. This is the principal on which interest is calculated. Therefore, P 545. The time is 1 month. Because the annual interest rate must be found and the time is given as 1 month, we write 1 month as

1 12

year, so t

1 . 12

To find the interest rate,

solve I Prt for r. Solution

Your solution

I Prt 8.72 545r

1 12

• Use the simple interest equation. • I 8.72, P 545, t

1 12

545 r 12 12 12 545 8.72 r 545 545 12 0.192 r 8.72

Example 9

You Try It 9

To make a certain color of blue, 4 oz of cyan must be contained in 1 gal of paint. What is the percent concentration of cyan in the paint?

The concentration of sugar in a certain breakfast cereal is 25%. If there are 2 oz of sugar contained in the cereal in a bowl, how many ounces of cereal are in the bowl?

Strategy

Your strategy

The cyan is given in ounces and the amount of paint is given in gallons. We must convert ounces to gallons or gallons to ounces. For this problem, we will convert gallons to ounces: 1 gal 128 oz. Solve Q Ar for r with Q 4 and A 128. Solution

Q Ar 4 128r 4 128r 128 128 0.03125 r

Your solution • Use the percent mixture equation. • Q 4, A 128

The percent concentration of cyan is 3.125%. Solutions on p. S4


The annual interest rate is 19.2%.


Objective E TA K E N O T E A car traveling in a circle at a constant speed of 45 mph is not in uniform motion because the direction of the car is always changing.

To solve uniform motion problems

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Any object that travels at a constant speed in a straight line is said to be in uniform motion. Uniform motion means that the speed and direction of an object do not change. For instance, a car traveling at a constant speed of 45 mph on a straight road is in uniform motion. The solution of a uniform motion problem is based on the uniform motion equation d rt, where d is the distance traveled, r is the rate of travel, and t is the time spent traveling. For instance, suppose a car travels at 50 mph for 3 h. Because the rate (50 mph) and time (3 h) are known, we can find the distance traveled by solving the equation d rt for d. d rt d 503 d 150

• r 50, t 3.

The car travels a distance of 150 mi.

HOW TO A jogger runs 3 mi in 45 min. What is the rate of the jogger in miles per hour? Strategy

Because the answer must be in miles per hour and the given time is in minutes, convert 45 min to hours. To find the rate of the jogger, solve the equation d rt for r.

Solution

t 45 min

45 3 h h 60 4

d rt

3r

3 4

3 r 4 4 4 3 r 3 3 3 4

• d 3, t

3 4

3

• Multiply each side of the equation 3 by the reciprocal of . 4


4r The rate of the jogger is 4 mph.

If two objects are moving in opposite directions, then the rate at which the distance between them is increasing is the sum of the speeds of the two objects. For instance, in the diagram below, two cars start from the same point and travel in opposite directions. The distance between them is changing at 70 mph.

30 mph

40 mph

30 + 40 = 70 mph

82


Similarly, if two objects are moving toward each other, the distance between them is decreasing at a rate that is equal to the sum of the speeds. The rate at which the two planes at the right are approaching one another is 800 mph.

450 mph 350 mph

800 mph

Two cars start from the same point and move in opposite HOW TO directions. The car moving west is traveling 45 mph, and the car moving east is traveling 60 mph. In how many hours will the cars be 210 mi apart? The distance is 210 mi. Therefore, d 210. The cars are moving in opposite directions, so the rate at which the distance between them is changing is the sum of the rates of each of the cars. The rate is 45 mph 60 mph 105 mph. Therefore, r 105. To find the time, solve the equation d rt for t.

Solution

d rt 210 105t 105t 210 105 105

60 mph

105 mph

• d 210, r 105 • Solve for t.

2t In 2 h, the cars will be 210 mi apart. If a motorboat is on a river that is flowing at a rate of 4 mph, then the boat will float down the river at a speed of 4 mph when the motor is not on. Now suppose the motor is turned on and the power adjusted so that the boat can travel 10 mph without the aid of the current. Then, if the boat is moving with the current, its effective speed is the speed of the boat using power plus the speed of the current: 10 mph 4 mph 14 mph. (See the figure below.)

4 mph

10 mph 14 mph

However, if the boat is moving against the current, the current slows the boat down, and the effective speed of the boat is the speed of the boat using power minus the speed of the current: 10 mph 4 mph 6 mph. (See the figure below.)

4 mph

10 mph 6 mph


45 mph

Strategy


83

There are other situations in which the preceding concepts may be applied. TA K E N O T E The term fts is an abbreviation for “feet per second.” Similarly, cms is “centimeters per second” and ms is “meters per second.”

An airline passenger is walking between two airline terminals HOW TO and decides to get on a moving sidewalk that is 150 ft long. If the passenger walks at a rate of 7 fts and the moving sidewalk moves at a rate of 9 fts, how long, in seconds, will it take the passenger to walk from one end of the moving sidewalk to the other? Round to the nearest thousandth. Strategy

The distance is 150 ft. Therefore, d 150. The passenger is traveling at 7 fts and the moving sidewalk is traveling at 9 fts. The rate of the passenger is the sum of the two rates, or 16 fts. Therefore, r 16. To find the time, solve the equation d rt for t.

Solution

d rt 150 16t 150 16t 16 16 9.375 t

• d 150, r 16 • Solve for t.

It will take 9.375 s for the passenger to travel the length of the moving sidewalk.

Example 10

You Try It 10

Two cyclists start at the same time at opposite ends of an 80-mile course. One cyclist is traveling 18 mph, and the second cyclist is traveling 14 mph. How long after they begin will they meet?

A plane that can normally travel at 250 mph in calm air is flying into a headwind of 25 mph. How far can the plane fly in 3 h?

Strategy

Your strategy


The distance is 80 mi. Therefore, d 80. The cyclists are moving toward each other, so the rate at which the distance between them is changing is the sum of the rates of each of the cyclists. The rate is 18 mph 14 mph 32 mph. Therefore, r 32. To find the time, solve the equation d rt for t.

Solution

d rt 80 32t 80 32t 32 32 2.5 t

Your solution • d 80, r 32 • Solve for t.

The cyclists will meet in 2.5 h.

Solution on p. S4

84


2.1 Exercises

1.

To determine whether a given number is a solution of an equation

What is the difference between an equation and an expression?

2.

Explain how to determine whether a given number is a solution of an equation.

3.

Is 4 a solution of 2x 8?

4. Is 3 a solution of y 4 7?

5. Is 1 a solution of 2b 1 3?

6.

Is 2 a solution of 3a 4 10?

7. Is 1 a solution of 4 2m 3?

8. Is 2 a solution of 7 3n 2?

9.

Is 5 a solution of 2x 5 3x?

10.

Is 4 a solution of 3y 4 2y?

11. Is 2 a solution of 3a 2 2 a?

12.

Is 3 a solution of z2 1 4 3z?

13.

Is 2 a solution of 2x2 1 4x 1?

14.

Is 1 a solution of y2 1 4y 3?

15.

Is 4 a solution of xx 1 x2 5?

16.

Is 3 a solution of 2aa 1 3a 3?

17.

Is a solution of

18.

Is

1 2

a solution of

19.

4y 1 3?

Objective B 21.

Is

2 5

1 4

8t 1 1?

a solution of

20.

5m 1 10m 3?

Is

3 4

a solution of

8x 1 12x 3?

To solve an equation of the form x a b

Can 0 ever be the solution of an equation? If so, give an example of an equation for which 0 is a solution.

Without solving x

22.

13 15

21 , 43

determine whether x is less than or greater than

21 . 43

Explain your answer.

For Exercises 23 to 64, solve and check. x57

24.

y39

25.

b 4 11

26.

z 6 10

27. 2 a 8

28.

5 x 12

29.

n 5 2

30.

x 6 5

31. b 7 7

32.

y 5 5

33.

z92

34.

n 11 1

35. 10 m 3

36.

8x5

37.

9 x 3

38.

10 y 4

23.


Objective A


39.

2x7

40.

8 n 1

41.

4 m 11

42.

6 y 5

43.

12 3 w

44.

9 5 x

45.

4 10 b

46.

7 2 x

48.

c

3 1 4 4

49.

x

50.

x

2 3 5 5

5 1 y 8 8

52.

4 2 a 9 9

53.

m

54.

b

1 1 6 3

2 3 3 4

56.

n

57.

58.

47. m

51.

55. x

2 1 3 3

2 2 5 3

1 1 2 2

1 1 2 4

5 1 x 6 4

59.

d 1.3619 2.0148

60.

w 2.932 4.801

61.

0.813 x 1.096

62.

1.926 t 1.042

63.

6.149 3.108 z

64.

5.237 2.014 x

Objective C 65. Copyright © Houghton Mifflin Company. All rights reserved.

85

1 2 c 4 3

To solve an equation of the form ax b

Without solving

15 x 41

23 , 25

determine whether x is less than or greater than 0. Explain your answer.

66.

Explain why multiplying each side of an equation by the reciprocal of the coefficient of the variable is the same as dividing each side of the equation by the coefficient.

For Exercises 67 to 110, solve and check. 67. 5x 15

68.

4y 28

69.

3b 0

70.

2a 0

71. 3x 6

72.

5m 20

73.

3x 27

74.

1 n 30 6

86


20

76. 18 2t

77. 0 5x

79.

49 7t

80.

x 2 3

81.

x 3 4

82.

83.

b 6 3

84.

3 y9 4

85.

2 x6 5

2 86. d 8 3

87.

3 m 12 5

88.

2n 0 3

89.

5x 0 6

90.

91.

3x 2 4

92.

3 3 c 4 5

93.

2 2 y 9 3

94.

6 3 b 7 4

95.

1 1 x 5 10

2 8 96. y 3 9

98.

3 a 4 8

97. 1

2 6 199. m 5 7

100. 5x 2x 14

102. 7d 4d 9

103.

10y 3y 21

78. 0 8a

2n 3

101.

3n 2n 20

104.

2x 5x 9

y 5 2

3z 9 8

105.

x 3.25 1.46

106.

z 7.88 2.95

107.

3.47a 7.1482

108.

2.31m 2.4255

109.

3.7x 7.881

110.

n 9.08 2.65


1 c 4

75.


Objective D

87

To solve application problems using the basic percent equation

111. Without solving an equation, indicate whether 40% of 80 is less than, equal to, or greater than 80% of 40.

112. Without whether

solving 1 % 4

an

equation,

indicate

of 80 is less than, equal to, or

greater than 25% of 80.

113. What is 35% of 80?

114.

What percent of 8 is 0.5?

115.

Find 1.2% of 60.

116. 8 is what percent of 5?

117.

125% of what is 80?

118.

What percent of 20 is 30?


120.


121.

Find 18% of 40.

122. What is 25% of 60?

123.

12% of what is 48?

124.

45% of what is 9?

125. What is 33 % of 27?

126.

Find 16 % of 30.

127.



129.


130.



1 3

1 4

2 3

1 2

131. 5 % of what is 21?

132.

37 % of what is 15?

133.

Find 15.4% of 50.

134. What is 18.5% of 46?

135.

1 is 0.5% of what?

136.

3 is 1.5% of what?

138.

1 2

139.

What is 250% of 12?

137.

3 4

% of what is 3?

% of what is 3?


140.

Education The stacked-bar graph at the right shows the number of people age 25 or older in the U.S. who have attained some type of degree beyond high school. a. In 2002, there were approximately 182.7 million people age 25 or older. What percent of the people age 25 or older had received an associate degree or a bachelor’s degree in 2002? Round to the nearest tenth of a percent. b. In 2000, there were approximately 177.5 million people age 25 or older. Was the percent of people in 2000 with a graduate degree less than or greater than the percent of people in 2002 with a graduate degree?

Associate degree Bachelor's degree Graduate degree 60 Number (in millions)

88

141. Chemistry Approximately 21% of air is oxygen. Using this estimate, determine how many liters of oxygen there are in a room containing 21,600 L of air. 142.

Record Sales According to Nielsen SoundScan, there were approximately 680 million record albums sold in the fourth quarter of 2002. This is about 39% of the total number of record albums sold that year. How many record albums were sold in 2002? Round to the nearest million.

143.

Income According to the U.S. Census Bureau, the median income fell 1.1% between two successive years. If the median income before the decline was $42,900, what was the median income the next year? Round to the nearest dollar.

15.9

50 40

28.5

17.1

30.3

30 20 11.5

12.3

2000

2002

10 0

2

145.

Sports According to www.superbowl.com, approximately 138.9 million people watched Super Bowl XXXVIII. What percent of the U.S. population watched Super Bowl XXXVIII? Use a figure of 290 million for the U.S. population. Round to the nearest tenth of a percent.

146.

Advertising Suppose 9.4 million people watch a 30second commercial for a new cellular phone during a broadcast of the TV show CSI. The cost of that commercial was approximately $470,000. If the cellular phone manufacturer makes a profit of $10 on every phone sold, what percent of the people watching the commercial would have to buy one phone for the company to recover the cost of the commercial? (Source: Nielsen Media Research/ San Diego Union)

147.

School Enrollment The circle graph at the right shows the percent of the U.S. population over 3 years old who are enrolled in school. To answer the question “How many people are enrolled in college or graduate school?” what additional piece of information is necessary?

College/ graduate school 22.5%

Nursery school/ preschool 6.2% Kindergarten 5.4%

Elementary school 44.2% High school 21.7%

Source: U.S. Census Bureau


144. Government To override a presidential veto, at least 66 % of the Sen3 ate must vote to override the veto. There are 100 senators in the Senate. What is the minimum number of votes needed to override a veto?


148. Investment If Kachina Caron invested $1200 in a simple interest account and earned $72 in 8 months, what is the annual interest rate? 149. Investment How much money must Andrea invest for two years in an account that earns an annual interest rate of 8% if she wants to earn $300 from the investment? 150. Investment Sal Boxer decided to divide a gift of $3000 into two different accounts. He placed $1000 in one account that earns an annual simple interest rate of 7.5%. The remaining money was placed in an account that earns an annual simple interest rate of 8.25%. How much interest will Sal earn from the two accounts after one year? 151. Investment If Americo invests $2500 at an 8% annual simple interest rate and Octavia invests $3000 at a 7% annual simple interest rate, which of the two will earn the greater amount of interest after one year? 152. Investment Makana invested $900 in a simple interest account that had an interest rate that was 1% more than that of her friend Marlys. If Marlys earned $51 after one year from an investment of $850, how much did Makana earn in one year? 153. Investment A $2000 investment at an annual simple interest rate of 6% earned as much interest after one year as another investment in an account that earns 8% simple interest. How much was invested at 8%? 154. Investment An investor placed $1000 in an account that earns 9% annual simple interest and $1000 in an account that earns 6% annual simple interest. If each investment is left in the account for the same period of time, is the interest rate on the combined investment less than 6%, between 6% and 9%, or greater than 9%?


155. Metallurgy The concentration of platinum in a necklace is 15%. If the necklace weighs 12 g, find the amount of platinum in the necklace. 156. Dye Mixtures A 250-milliliter solution of a fabric dye contains 5 ml of hydrogen peroxide. What is the percent concentration of the hydrogen peroxide? 157. Fabric Mixtures A carpet is made of a blend of wool and other fibers. If the concentration of wool in the carpet is 75% and the carpet weighs 175 lb, how much wool is in the carpet? 158. Juice Mixtures Apple Dan’s 32-ounce apple-flavored fruit drink contains 8 oz of apple juice. A 40-ounce generic brand of an apple-flavored fruit drink contains 9 oz of apple juice. Which of the two brands has the greater concentration of apple juice?

89

90


159. Food Mixtures Bakers use simple syrup in many of their recipes. Simple syrup is made by combining 500 g of sugar with 500 g of water and mixing it well until the sugar dissolves. What is the percent concentration of sugar in the simple syrup?

160. Pharmacology A pharmacist has 50 g of a topical cream that contains 75% glycerine. How many grams of the cream is not glycerine?

161. Chemistry A chemist has 100 ml of a solution that is 9% acetic acid. If the chemist adds 50 ml of pure water to this solution, what is the percent concentration of the resulting mixture?

162. Chemistry A 500-gram salt-and-water solution contains 50 g of salt. This mixture is left in the open air and 100 g of water evaporates from the solution. What is the percent concentration of salt in the remaining solution?

Objective E


163. As part of the training program for the Boston Marathon, a runner wants to build endurance by running at a rate of 9 mph for 20 min. How far will the runner travel in that time period?

164. It takes a hospital dietician 40 min to drive from home to the hospital, a distance of 20 mi. What is the dietician’s average rate of speed?

165. Marcella leaves home at 9:00 A.M. and drives to school, arriving at 9:45 A.M. If the distance between home and school is 27 mi, what is Marcella’s average rate of speed?

167. Palmer’s average running speed is 3 kilometers per hour faster than his walking speed. If Palmer can run around a 30-kilometer course in 2 h, how many hours would it take for Palmer to walk the same course?

168. A shopping mall has a moving sidewalk that takes shoppers from the shopping area to the parking garage, a distance of 250 ft. If your normal walking rate is 5 fts and the moving sidewalk is traveling at 3 fts, how many seconds would it take for you to walk from one end of the moving sidewalk to the other end?


166. The Ride for Health Bicycle Club has chosen a 36-mile course for this Saturday’s ride. If the riders plan on averaging 12 mph while they are riding, and they have a 1-hour lunch break planned, how long will it take them to complete the trip?


169. Two joggers start at the same time from opposite ends of an 8-mile jogging trail and begin running toward each other. One jogger is running at the rate of 5 mph, and the other jogger is running at a rate of 7 mph. How long, in minutes, after they start will the two joggers meet? 170. Two cyclists start from the same point at the same time and move in opposite directions. One cyclist is traveling at 8 mph, and the other cyclist is traveling at 9 mph. After 30 min, how far apart are the two cyclists? 171. Petra and Celine can paddle their canoe at a rate of 10 mph in calm water. How long will it take them to travel 4 mi against the 2 mph current of the river? 172. At 8:00 A.M., a train leaves a station and travels at a rate of 45 mph. At 9:00 A.M., a second train leaves the same station on the same track and travels in the direction of the first train at a speed of 60 mph. At 10:00 A.M., how far apart are the two trains?

APPLYING THE CONCEPTS 173. Solve the equation ax b for x. Is the solution you have written valid for all real numbers a and b?


174. Solve.

a.

3 5 1 x

b.

2 2 1 y

c.

3x 2x 2 3

175. a. Make up an equation of the form x a b that has 2 as a solution. b. Make up an equation of the form ax b that has 1 as a solution.

176.

Write out the steps for solving the equation

1 2

x 3. Identify

each Property of Real Numbers or Property of Equations as you use it. 177.

In your own words, state the Addition Property of Equations and the Multiplication Property of Equations.

178. If a quantity increases by 100%, how many times its original value is its new value?

91

92


2.2

General Equations ax b c

Objective A

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In solving an equation of the form ax b c, the goal is to rewrite the equation in the form variable constant. This requires the application of both the Addition and the Multiplication Properties of Equations. 3 x 2 11 4 The goal is to write the equation in the form variable constant. HOW TO

Solve:

3 x 2 11 4

Check:

3 x 2 11 4 3 12 2 11 4 9 2 11 11 11 A true equation

3 x 2 2 11 2 4 3 x 9 4

• Add 2 to each side of the equation. • Simplify.

4 3 4 x 9 3 4 3 x 12

4 3

• Multiply each side of the equation by . • The equation is in the form variable constant .

The solution is 12. Here is an example of solving an equation that contains more than one fraction. HOW TO

Solve:

2 1 3 x 3 2 4

2 1 3 x 3 2 4 2 1 1 3 1 x 3 2 2 4 2 2 1 x 3 4

3 2 3 1 x 2 3 2 4 x

3 8

1 2

• Subtract from each side of the equation. • Simplify. 3 2

• Multiply each side of the equation by , 2 3

the reciprocal of .

3 8

The solution is . It may be easier to solve an equation containing two or more fractions by multiplying each side of the equation by the least common multiple (LCM) of the denominators. For the equation above, the LCM of 3, 2, and 4 is 12. The LCM has the property that 3, 2, and 4 will divide evenly into it. Therefore, if both sides of the equation are multiplied by 12, the denominators will divide evenly into 12. The result is an equation that does not contain any fractions. Multiplying each side of an equation that contains fractions by the LCM of the denominators is called clearing denominators. It is an alternative method, as we show in the next example, of solving an equation that contains fractions.


TA K E N O T E

Section 2.2 / General Equations

HOW TO

Solve:

93

2 1 3 x 3 2 4

2 1 3 x 3 2 4

2 1 x 3 2

12

3 4

2 1 x 12 3 2

12

3 4

12

TA K E N O T E Observe that after we multiply by the LCM and simplify, the equation no longer contains fractions. Also note that this is the same equation solved on the previous page.

12

8x 6 9 8x 6 6 9 6

• Multiply each side of the equation by 12, the LCM of 3, 2, and 4.

• Use the Distributive Property. • Simplify. • Subtract 6 from each side of the equation.

8x 3 3 8x 8 8 x

• Divide each side of the equation by 8.

3 8

3 8

The solution is . Note that both methods give exactly the same solution. You may use either method to solve an equation containing fractions.

Example 1

Solve: 3x 7 5

Solution

You Try It 1

Solve: 5x 7 10

Your solution

3x 7 5 3x 7 7 5 7 3x 2 3x 2 3 3 2 x 3

• Add 7 to each side.

• Divide each side by 3.

2 3


The solution is .

Example 2

Solve: 5 9 2x

Solution

5 9 2x 5 9 9 9 2x 4 2x 4 2x 2 2 2x

You Try It 2

Solve: 2 11 3x

Your solution • Subtract 9 from each side. • Divide each side by 2.

The solution is 2. Solutions on p. S4

94


Example 3

Solve:

2 x 3 3 2 4

You Try It 3

Solution

5 2x 5 8 3 4

Your solution

2 x 3 3 2 4 2 2 x 3 3 3 2 4 x 1 2 12 x 2 2 2 1 x 6

2 3

The solution is

Example 4

Solve:

1 12

• Subtract

2 from each side. 3

• Multiply each side by 2.

1 . 6

4 1 3 x by first 5 2 4 clearing denominators. Solve

Solution

You Try It 4

2 7 x 3 by first 3 2 clearing denominators. Solve

Your solution


20

4 1 3 x 20 20 5 2 4 16x 10 15 16x 10 10 15 10 16x 25 25 16x 16 16 25 x 16

The solution is

• Multiply each side by 20. • Use the Distributive Property. • Add 10 to each side.


25 . 16

Solutions on pp. S4– S5


1 3 4 x 5 2 4 1 4 3 20 x 20 5 2 4


Example 5

You Try It 5

Solve: 2x 4 5x 10

Solve: x 5 4x 25

Solution

Your solution

2x 4 5x 10 3x 4 10 3x 4 4 10 4 3x 6 3x 6 3 3 x 2

95

• Combine like terms. • Subtract 4 from each side of the equation.


The solution is 2.

Solution on p. S5

Objective B

Study

Tip


Have you considered joining a study group? Getting together regularly with other students in the class to go over material and quiz each other can be very beneficial. See AIM for Success, page xxvi.

To solve an equation of the form ax b cx d

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In solving an equation of the form ax b cx d, the goal is to rewrite the equation in the form variable constant. Begin by rewriting the equation so that there is only one variable term in the equation. Then rewrite the equation so that there is only one constant term. HOW TO

Solve: 2x 3 5x 9

2x 3 5x 9 2x 5x 3 5x 5x 9

• Subtract 5x from each side of the equation.

3x 3 9

• Simplify. There is only one variable term.

3x 3 3 9 3

• Subtract 3 from each side of the equation.

3x 12

• Simplify. There is only one constant term.

3x 12 3 3


x4

• The equation is in the form variable constant .

The solution is 4. You should verify this by checking this solution.

96


Example 6

Solve: 4x 5 8x 7

Solution

You Try It 6

Solve: 5x 4 6 10x

Your solution

4x 5 8x 7 4x 8x 5 8x 8x 7 • Subtract 8x from each side.

4x 5 7 4x 5 5 7 5

• Add 5 to each side.

4x 2 4x 2 4 4 x


1 2 1 2

The solution is .

Example 7

Solve: 3x 4 5x 2 4x

Solution

You Try It 7

Solve: 5x 10 3x 6 4x

Your solution

3x 4 5x 2 4x 2x 4 2 4x


2x 4x 4 2 4x 4x

• Add 4x to each side.

2x 4 2 2x 4 4 2 4

• Subtract 4 from each side.

2x 2 2 2


x 1 The solution is 1.

Solutions on p. S5


2x 2

97


Objective C

To solve an equation containing parentheses

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When an equation contains parentheses, one of the steps in solving the equation requires the use of the Distributive Property. The Distributive Property is used to remove parentheses from a variable expression.

HOW TO

Solve: 4 52x 3 34x 1

4 52x 3 34x 1 4 10x 15 12x 3

• Use the Distributive Property. Then simplify.

10x 11 12x 3 10x 12x 11 12x 12x 3 2x 11 3 2x 11 11 3 11

• Subtract 12x from each side of the equation. • Simplify. • Add 11 to each side of the equation.

2x 8

• Simplify.

2x 8 2 2


x 4

• The equation is in the form variable constant .

The solution is 4. You should verify this by checking this solution.

In the next example, we solve an equation with parentheses and decimals.

HOW TO

Solve: 16 0.55x 0.75x 20


16 0.55x 0.75x 20 16 0.55x 0.75x 15 16 0.55x 0.75x 0.75x 0.75x 15

• Use the Distributive Property. • Subtract 0.75x from each side of the equation.

16 0.20x 15 16 16 0.20x 15 16

• Simplify. • Subtract 16 from each side of the equation.

0.20x 1

• Simplify.

0.20x 1 0.20 0.20

• Divide each side of the equation by

x5 The solution is 5.

0.20.

• The equation is in the form variable constant.

98


Example 8

You Try It 8

Solve: 3x 42 x 3x 2 4

Solve: 5x 43 2x 23x 2 6

Solution

Your solution

3x 42 x 3x 2 4 3x 8 4x 3x 6 4 7x 8 3x 10 7x 3x 8 3x 3x 10 4x 8 10 4x 8 8 10 8

• Distributive Property • Subtract 3x. • Add 8.

4x 2 4x 2 4 4 x

• Divide by 4.

1 2

1 2

The solution is .

Example 9

You Try It 9

Solve: 32 42x 1 4x 10

Solve: 23x 52x 3 3x 8

Solution

Your solution

18 24x 4x 10 18 24x 4x 4x 4x 10 18 28x 10 18 18 28x 10 18 28x 28 28x 28 28 28

• Distributive Property • Distributive Property • Subtract 4x. • Subtract 18.

• Divide by 28.

x1 The solution is 1.

Solutions on p. S5


32 42x 1 4x 10 32 8x 4 4x 10 36 8x 4x 10


Objective D TA K E N O T E 90 lb

60 lb 4

6

10 ft

This system balances because F1 x F2 d x 606 9010 6 606 904 360 360

To solve application problems using formulas A lever system is shown at the right. It consists of a lever, or bar; a fulcrum; and two forces, F1 and F2. The distance d represents the length of the lever, x represents the distance from F1 to the fulcrum, and d x represents the distance from F2 to the fulcrum.

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F1

99

F2 d−x

x

Lever Fulcrum d

A principle of physics states that when the lever system balances, F1 x F2 d x.

Example 10

You Try It 10

A lever is 15 ft long. A force of 50 lb is applied to one end of the lever, and a force of 100 lb is applied to the other end. Where is the fulcrum located when the system balances?

A lever is 25 ft long. A force of 45 lb is applied to one end of the lever, and a force of 80 lb is applied to the other end. Where is the fulcrum located when the system balances?

Strategy

Your strategy

Make a drawing.

100 lb 50 lb

x

d–x


d

Given: F1 50 F2 100 d 15 Unknown: x Solution

Your solution

F1 x F2 d x 50x 10015 x 50x 1500 100x 50x 100x 1500 100x 100x 150x 1500 150x 1500 150 150 x 10 The fulcrum is 10 ft from the 50-pound force. Solution on p. S5

100



To solve an equation of the form ax b c

For Exercises 1 to 80, solve and check. 3x 1 10

5. 5 4x 9

9. 4 3w 2

2. 4y 3 11

3. 2a 5 7

4. 5m 6 9

6. 2 5b 12

7. 2x 5 11

8. 3n 7 19

10. 5 6x 13

11. 8 3t 2

12. 12 5x 7

13. 4a 20 0

14.

3y 9 0

15.

6 2b 0

16.

10 5m 0

17. 2x 5 7

18.

5d 3 12

19.

12x 30 6

20.

13 11y 9

21. 2 7 5a

22.

3 11 4n

23.

35 6b 1

24.

8x 3 29

25. 3m 21 0

26.

5x 30 0

27.

4y 15 15

28.

3x 19 19

29. 9 4x 6

30.

3t 2 0

31.

9x 4 0

32.

7 8z 0

33. 1 3x 0

34.

9d 10 7

35.

12w 11 5

36.

6y 5 7

37. 8b 3 9

38.

5 6m 2

39.

7 9a 4

40.

9 12c 5


1.


42.

2y

1 7 3 3

43.

4a

3 19 4 4

44.

2n

5 13 6 6

46.

5y

3 3 7 7

47.

9x

4 4 5 5

48.

8 7d 1

49. 8 10x 5

50.

4 7 2w

51.

7 9 5a

52.

8t 13 3

53. 12x 19 3

54.

6y 5 13

55.

4x 3 9

56.

1 a31 2

58.

2 y46 5

59.

3 n 7 13 4

60.

62.

x 61 4

63.

y 23 5

64.

2x 15 3

67.

1 2 1 x 2 3 4

68.

3 3 19 x 4 5 20

71.

4 2x 11 27 9 3

72.

37 7 5x 24 8 6

76.

6a 3 2a 11

80.

2x 6x 1 9

41. 10 18x 7

45. 3x

57.

1 m15 3

61.

3 b 4 10 8

65.

2 5 1 x 3 6 3

66.

5 2 1 x 4 3 4

69.

5 3x 3 2 6 8

70.

2x 4 5

74.

5

4c 8 7

75.

7

78.

7x 4 2x 6

79.

11z 3 7z 9

73. 7 Copyright © Houghton Mifflin Company. All rights reserved.

101

77. 5y 9 2y 23

1 5 5x 4 12 6

81. Solve 3x 4y 13 when y 2.

83.

Solve 4x 3y 9 when x 0.

5 y9 9

3 13 4 4

2 x17 3

82.

Solve 2x 3y 8 when y 0.

84.

Solve 5x 2y 3 when x 3.

102


Objective B

To solve an equation of the form ax b cx d

For Exercises 85 to 111, solve and check. 86.

6y 2 y 17

87. 5x 4 2x 5

88. 13b 1 4b 19

89. 15x 2 4x 13

90. 7a 5 2a 20

91. 3x 1 11 2x

92. n 2 6 3n

93. 2x 3 11 2x

94. 4y 2 16 3y

95. 2b 3 5b 12

96. m 4 3m 8

97. 4y 8 y 8

98. 5a 7 2a 7

99. 6 5x 8 3x

100. 10 4n 16 n

101.

5 7x 11 9x

102.

3 2y 15 4y

103. 2x 4 6x

104.

2b 10 7b

105.

8m 3m 20

106. 9y 5y 16

107.

8b 5 5b 7

108.

6y 1 2y 2

109. 7x 8 x 3

110.

2y 7 1 2y

111.

2m 1 6m 5

112. If 5x 3x 8, evaluate 4x 2.

113.

If 7x 3 5x 7, evaluate 3x 2.

114. If 2 6a 5 3a, evaluate 4a2 2a 1.

115.

If 1 5c 4 4c, evaluate 3c2 4c 2.

116. If 2y 3 5 4y, evaluate 6y 7.

117.

If 3z 1 1 5z, evaluate 3z2 7z 8.


85. 8x 5 4x 13


Objective C

103

To solve an equation containing parentheses


For Exercises 118 to 138, solve and check. 118. 5x 2x 1 23

119.

6y 22y 3 16

120.

9n 32n 1 15

121. 12x 24x 6 28

122.

7a 3a 4 12

123.

9m 42m 3 11

124. 53 2y 4y 3

125.

41 3x 7x 9

126.

5y 3 7 4y 2

127. 0.22x 6 0.2x 1.8

128.

0.054 x 0.1x 0.32

129.

0.3x 0.3x 10 300

130. 2a 5 43a 1 2

131.

5 9 6x 2x 2

132.

7 5 8x 4x 3

133. 32 4y 1 32y 8

134.

52 2x 4 25 3x

135. 3a 22 3a 1 23a 4

136.

5 31 22x 3 6x 5

137. 24 3b 2 5 23b 6

138.

4x 22x 3 1 2x 3

139. If 4 3a 7 22a 5 , evaluate a2 7a .

140.

If 9 5x 12 6x 7, evaluate x2 3x 2.

142.

If 3n 7 52n 7, evaluate

141. If 2z 5 34z 5, evaluate

z2 . z2

n2 . 2n 6

104


Objective D

To solve application problems using formulas

Physics For Exercises 143 to 149, solve. Use the lever system equation F1x F2d x.

F2 100 lb 2 ft

143. A lever 10 ft long is used to move a 100-pound rock. The fulcrum is placed 2 ft from the rock. What force must be applied to the other end of the lever to move the rock?

144. An adult and a child are on a seesaw 14 ft long. The adult weighs 175 lb and the child weighs 70 lb. How many feet from the child must the fulcrum be placed so that the seesaw balances?

145. Two people are sitting 15 ft apart on a seesaw. One person weighs 180 lb. The second person weighs 120 lb. How far from the 180-pound person should the fulcrum be placed so that the seesaw balances?

120 lb

180 lb 15 − x

x

15 ft

146. Two children are sitting on a seesaw that is 12 ft long. One child weighs 60 lb. The other child weighs 90 lb. How far from the 90-pound child should the fulcrum be placed so that the seesaw balances? 30

147. In preparation for a stunt, two acrobats are standing on a plank 18 ft long. One acrobat weighs 128 lb and the second acrobat weighs 160 lb. How far from the 128-pound acrobat must the fulcrum be placed so that the acrobats are balanced on the plank?

lb

0.15 in.

F1

9i

n.

148. A screwdriver 9 in. long is used as a lever to open a can of paint. The tip of the screwdriver is placed under the lip of the can with the fulcrum 0.15 in. from the lip. A force of 30 lb is applied to the other end of the screwdriver. Find the force on the lip of the can. F2

Business To determine the break-even point, or the number of units that must be sold so that no profit or loss occurs, an economist uses the formula Px Cx F, where P is the selling price per unit, x is the number of units that must be sold to break even, C is the cost to make each unit, and F is the fixed cost. Use this equation for Exercises 150 to 155. 150. A business analyst has determined that the selling price per unit for a laser printer is $1600. The cost to make one laser printer is $950, and the fixed cost is $211,250. Find the break-even point.

(8 –

1.5)

ft

15 1.5 f 0 lb t Copyright © Houghton Mifflin Company. All rights reserved.

149. A metal bar 8 ft long is used to move a 150-pound rock. The fulcrum is placed 1.5 ft from the rock. What minimum force must be applied to the other end of the bar to move the rock? Round to the nearest tenth.


151. A business analyst has determined that the selling price per unit for a gas barbecue is $325. The cost to make one gas barbecue is $175, and the fixed cost is $39,000. Find the break-even point.

152. A manufacturer of thermostats determines that the cost per unit for a programmable thermostat is $38 and that the fixed cost is $24,400. The selling price for the thermostat is $99. Find the break-even point.

153. A manufacturing engineer determines that the cost per unit for a desk lamp is $12 and that the fixed cost is $19,240. The selling price for the desk lamp is $49. Find the break-even point.

154. A manufacturing engineer determines the cost to make one compact disc to be $3.35 and the fixed cost to be $6180. The selling price for each compact disc is $8.50. Find the number of compact discs that must be sold to break even.

155. To manufacture a softball bat requires two steps. The first step is to cut a rough shape. The second step is to sand the bat to its final form. The cost to rough-shape a bat is $.45, and the cost to sand a bat to final form is $1.05. The total fixed cost for the two steps is $16,500. How many softball bats must be sold at a price of $7.00 to break even?

APPLYING THE CONCEPTS 156. Write an equation of the form ax b cx d that has 4 as the solution. For Exercises 157 to 160, solve. If the equation has no solution, write “no solution.” 157. 32x 1 6x 4 9


158. 73x 6 43 5x 13 x

159.

1 1 25 10a 4 12a 15 14 5 3

160. 5m 23 m 324 m 5 161.

The equation x x 1 has no solution, whereas the solution of the equation 2x 3 3 is zero. Is there a difference between no solution and a solution of zero? Explain your answer.

105

106


2.3 Objective A

Translating Sentences into Equations To solve integer problems

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An equation states that two mathematical expressions are equal. Therefore, to translate a sentence into an equation requires recognition of the words or phrases that mean “equals.” Some of these phrases are listed below.  equals  is  is equal to  translate to amounts to   represents 

Once the sentence is translated into an equation, the equation can be solved by rewriting the equation in the form variable constant. HOW TO and solve.

Translate “five less than a number is thirteen” into an equation

The unknown number: n Five less than a number

is

thirteen

n5

13

TA K E N O T E You can check the solution to a translation problem. Check: 5 less than 18 is 13 18 5 13 13 13

• Assign a variable to the unknown number. • Find two verbal expressions for the same value.

• Write a mathematical expression for each verbal expression. Write the equals sign.

n 5 5 13 5

• Solve the equation.

n 18

Recall that the integers are the numbers {. . . , 4, 3, 2, 1, 0, 1, 2, 3, 4, . . .}. An even integer is an integer that is divisible by 2. Examples of even integers are 8, 0, and 22. An odd integer is an integer that is not divisible by 2. Examples of odd integers are 17, 1, and 39.

TA K E N O T E Both consecutive even and consecutive odd integers are represented using n, n 2, n 4, ....

Consecutive integers are integers that follow one another in order. Examples of consecutive integers are shown at the right. (Assume that the variable n represents an integer.)

11, 12, 13 8, 7, 6 n, n 1, n 2

Examples of consecutive even integers are shown at the right. (Assume that the variable n represents an even integer.)

24, 26, 28 10, 8, 6 n, n 2, n 4

Examples of consecutive odd integers are shown at the right. (Assume that the variable n represents an odd integer.)

19, 21, 23 1, 1, 3 n, n 2, n 4


The number is 18.

Section 2.3 / Translating Sentences into Equations

107

HOW TO The sum of three consecutive odd integers is forty-five. Find the integers. Strategy

• First odd integer: n • Represent three consecutive odd integers. Second odd integer: n 2 Third odd integer: n 4 • The sum of the three odd integers is 45. Solution

n n 2 n 4 45

• Write an equation.

3n 6 45

• Solve the equation.

3n 39 n 13

• The first odd integer is 13.

n 2 13 2 15

• Find the second odd integer.

n 4 13 4 17

• Find the third odd integer.

The three consecutive odd integers are 13, 15, and 17.

Example 1

You Try It 1

The sum of two numbers is sixteen. The difference between four times the smaller number and two is two more than twice the larger number. Find the two numbers.

The sum of two numbers is twelve. The total of three times the smaller number and six amounts to seven less than the product of four and the larger number. Find the two numbers.

Solution

Your solution

The smaller number: n The larger number: 16 n


The difference between four times the smaller and two

is

two more than twice the larger

4n 2 216 n 2 4n 2 32 2n 2 4n 2 34 2n 4n 2n 2 34 2n 2n 6n 2 34 6n 2 2 34 2 6n 36 6n 36 6 6 n6 16 n 16 6 10 The smaller number is 6. The larger number is 10. Solution on p. S6

108


Example 2

You Try It 2

Find three consecutive even integers such that three times the second equals four more than the sum of the first and third.

Find three consecutive integers whose sum is negative six.

Strategy

Your strategy

• First even integer: n Second even integer: n 2 Third even integer: n 4 • Three times the second equals four more than the sum of the first and third. Your solution

Solution

3n 2 n n 4 4 3n 6 2n 8 3n 2n 6 2n 2n 8 n68 n2 n2224 n4246 The three integers are 2, 4, and 6. Solution on p. S6

To translate a sentence into an equation and solve

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Example 3

You Try It 3

A wallpaper hanger charges a fee of $25 plus $12 for each roll of wallpaper used in a room. If the total charge for hanging wallpaper is $97, how many rolls of wallpaper were used?

The fee charged by a ticketing agency for a concert is $3.50 plus $17.50 for each ticket purchased. If your total charge for tickets is $161, how many tickets are you purchasing?

Strategy

Your strategy

To find the number of rolls of wallpaper used, write and solve an equation using n to represent the number of rolls of wallpaper used. Solution

Your solution

$25 plus $12 for each roll of wallpaper

is

$97

25 12n 97 12n 72 12n 72 12 12 n6 Six rolls of wallpaper were used. Solution on p. S6


Objective B


109

Example 4

You Try It 4

A board 20 ft long is cut into two pieces. Five times the length of the shorter piece is 2 ft more than twice the length of the longer piece. Find the length of each piece.

A wire 22 in. long is cut into two pieces. The length of the longer piece is 4 in. more than twice the length of the shorter piece. Find the length of each piece.

Strategy

Your strategy

Let x represent the length of the shorter piece. Then 20 x represents the length of the longer piece.

x

20

ft

20

–x

Make a drawing.

To find the lengths, write and solve an equation using x to represent the length of the shorter piece and 20 x to represent the length of the longer piece.

Your solution

Solution


Five times the length of the shorter piece

ft more than twice is 2the length of the longer

5x 220 x 2 5x 40 2x 2 5x 42 2x 5x 2x 42 2x 2x 7x 42 7x 42 7 7 x6 20 x 20 6 14 The length of the shorter piece is 6 ft. The length of the longer piece is 14 ft.

Solution on p. S6

110



To solve integer problems

1.

The difference between a number and fifteen is seven. Find the number.

2.

The sum of five and a number is three. Find the number.

3.

The product of seven and a number is negative twenty-one. Find the number.

4.

The quotient of a number and four is two. Find the number.

5.

The difference between nine and a number is seven. Find the number.

6.

Three-fifths of a number is negative thirty. Find the number.

7.

The difference between five and twice a number is one. Find the number.

8.

Four more than three times a number is thirteen. Find the number.

9.

The sum of twice a number and five is fifteen. Find the number.

10. The difference between nine times a number and six is twelve. Find the number.

11. Six less than four times a number is twenty-two. Find the number.

12. Four times the sum of twice a number and three is twelve. Find the number.

13. Three times the difference between four times a number and seven is fifteen. Find the number.

14. Twice the difference between a number and twenty-five is three times the number. Find the number.

15. The sum of two numbers is twenty. Three times the smaller is equal to two times the larger. Find the two numbers.

16. The sum of two numbers is fifteen. One less than three times the smaller is equal to the larger. Find the two numbers.

17. The sum of two numbers is fourteen. The difference between two times the smaller and the larger is one. Find the two numbers.

18. The sum of two numbers is eighteen. The total of three times the smaller and twice the larger is forty-four. Find the two numbers.

19. The sum of three consecutive odd integers is fifty-one. Find the integers.

20. Find three consecutive even integers whose sum is negative eighteen.

21. Find three consecutive odd integers such that three times the middle integer is one more than the sum of the first and third.

22. Twice the smallest of three consecutive odd integers is seven more than the largest. Find the integers.

23. Find two consecutive even integers such that three times the first equals twice the second.

24. Find two consecutive even integers such that four times the first is three times the second.

25. Seven times the first of two consecutive odd integers is five times the second. Find the integers.

26. Find three consecutive even integers such that three times the middle integer is four more than the sum of the first and third.


For Exercises 1 to 18, translate into an equation and solve.


Objective B

To translate a sentence into an equation and solve

27. Computer Science The processor speed of a personal computer is 3.2 gigahertz (GHz). This is three-fourths the processor speed of a newer model personal computer. Find the processor speed of the newer personal computer.

28. Computer Science The storage capacity of a hard-disk drive is 60 gigabytes. This is one-fourth the storage capacity of a second hard-disk drive. Find the storage capacity of the second hard-disk drive.

29. Geometry An isosceles triangle has two sides of equal length. The length of the third side is 1 ft less than twice the length of an equal side. Find the length of each side when the perimeter is 23 ft.

30. Geometry An isosceles triangle has two sides of equal length. The length of one of the equal sides is two more than 3 times the length of the third side. If the perimeter is 46 m, find the length of each side.

31. Union Dues A union charges monthly dues of $4.00 plus $.15 for each hour worked during the month. A union member’s dues for March were $29.20. How many hours did the union member work during the month of March?

32. Technical Support A technical information hotline charges a customer $15.00 plus $2.00 per minute to answer questions about software. How many minutes did a customer who received a bill for $37 use this service?


33. Construction The total cost to paint the inside of a house was $1346. This cost included $125 for materials and $33 per hour for labor. How many hours of labor were required to paint the inside of the house?

34. Telecommunications The cellular phone service for a business executive is $35 per month plus $.40 per minute of phone use. In a month when the executive’s cellular phone bill was $99.80, how many minutes did the executive use the phone?

35. Computer Science A computer screen consists of tiny dots of light called pixels. In a certain graphics mode, there are 1280 horizontal pixels. This is 768 less than twice the number of vertical pixels. Find the number of vertical pixels.

111

112


36. Energy The cost of electricity in a certain city is $.08 for each of the first 300 kWh (kilowatt-hours) and $.13 for each kilowatt-hour over 300 kWh. Find the number of kilowatt-hours used by a family with a $51.95 electric bill.

37. Geometry The distance around a rectangular path is 42 m. The length of the path is 3 m less than twice the width. Find the length and width of the path.

38. Geometry The fence around a rectangular vegetable garden is 64 ft. The length of the garden is 20 ft. Find the width of the garden.

39. Carpentry A 12-foot board is cut into two pieces. Twice the length of the shorter piece is 3 ft less than the length of the longer piece. Find the length of each piece.

40. Sports A 14-yard fishing line is cut into two pieces. Three times the length of the longer piece is four times the length of the shorter piece. Find the length of each piece.

41. Education Seven thousand dollars is divided into two scholarships. Twice the amount of the smaller scholarship is $1000 less than the larger scholarship. What is the amount of the larger scholarship?

42. Investing An investment of $10,000 is divided into two accounts, one for stocks and one for mutual funds. The value of the stock account is $2000 less than twice the value of the mutual funds account. Find the amount in each account.

43.

Make up two word problems; one that requires solving the equation 6x 123 and one that requires solving the equation 8x 100 300.

44.

A formula is an equation that relates variables in a known way. Find two examples of formulas that are used in your college major. Explain what each of the variables represents.

45.

It is always important to check the answer to an application problem to be sure the answer makes sense. Consider the following problem. A 4-quart mixture of fruit juices is made from apple juice and cranberry juice. There are 6 more quarts of apple juice than of cranberry juice. Write and solve an equation for the number of quarts of each juice used. Does the answer to this question make sense? Explain.



Section 2.4 / Mixture and Uniform Motion Problems

2.4 Objective A

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Mixture and Uniform Motion Problems To solve value mixture problems

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A value mixture problem involves combining ingredients that have different prices into a single blend. For example, a coffee merchant may blend two types of coffee into a single blend, or a candy manufacturer may combine two types of candy to sell as a variety pack. TA K E N O T E The equation AC V is used to find the value of an ingredient. For example, the value of 4 lb of cashews costing $6 per pound is AC V 4 $6 V $24 V

The solution of a value mixture problem is based on the value mixture equation AC V, where A is the amount of an ingredient, C is the cost per unit of the ingredient, and V is the value of the ingredient. HOW TO A coffee merchant wants to make 6 lb of a blend of coffee costing $5 per pound. The blend is made using a $6-per-pound grade and a $3-per-pound grade of coffee. How many pounds of each of these grades should be used?

Strategy for Solving a Value Mixture Problem 1. For each ingredient in the mixture, write a numerical or variable expression for the

amount of the ingredient used, the unit cost of the ingredient, and the value of the amount used. For the blend, write a numerical or variable expression for the amount, the unit cost of the blend, and the value of the amount. The results can be recorded in a table.

Amount of $3 coffee: 6 x Amount of $6 coffee: x

The sum of the amounts is 6 lb.


TA K E N O T E Use the information given in the problem to fill in the amount and unit cost columns of the table. Fill in the value column by multiplying the two expressions you wrote in each row. Use the expressions in the last column to write the equation.

$6 grade $3 grade $5 blend

Amount, A

Unit Cost, C

Value, V

x 6x 6

6 3 5

6x 36 x 56

2. Determine how the values of the ingredients are related. Use the fact that the sum of

the values of all the ingredients is equal to the value of the blend.

The sum of the values of the $6 grade and the $3 grade is equal to the value of the $5 blend. 6x 36 x 56 6x 18 3x 30 3x 18 30 3x 12 x4 6x642

• Find the amount of the $3 grade coffee.

The merchant must use 4 lb of the $6 coffee and 2 lb of the $3 coffee.

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Example 1

You Try It 1

How many ounces of a silver alloy that costs $4 an ounce must be mixed with 10 oz of an alloy that costs $6 an ounce to make a mixture that costs $4.32 an ounce?

A gardener has 20 lb of a lawn fertilizer that costs $.80 per pound. How many pounds of a fertilizer that costs $.55 per pound should be mixed with this 20 lb of lawn fertilizer to produce a mixture that costs $.75 per pound?

Strategy

Your strategy

x oz $4/oz

10 oz $ 6/o z

• Ounces of $4 alloy: x

$4 alloy $6 alloy $4.32 mixture

Amount

Cost

Value

x 10 10 x

4 6 4.32

4x 610 4.3210 x

• The sum of the values before mixing equals the value after mixing.

Your solution

4x 610 4.3210 x 4x 60 43.2 4.32x 0.32x 60 43.2 0.32x 16.8 x 52.5 52.5 oz of the $4 silver alloy must be used.

Solution on p. S6


Solution


Objective B

To solve percent mixture problems

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Recall from Section 2.1 that a percent mixture problem can be solved using the equation Ar Q, where A is the amount of a solution, r is the percent concentration of a substance in the solution, and Q is the quantity of the substance in the solution. Ar Q 5000.04 Q 20 Q

For example, a 500-milliliter bottle is filled with a 4% solution of hydrogen peroxide. The bottle contains 20 ml of hydrogen peroxide.

HOW TO How many gallons of a 20% salt solution must be mixed with 6 gal of a 30% salt solution to make a 22% salt solution?

Strategy for Solving a Percent Mixture Problem 1. For each solution, write a numerical or variable expression for the amount of

solution, the percent concentration, and the quantity of the substance in the solution. The results can be recorded in a table.

The unknown quantity of 20% solution: x

TA K E N O T E


Use the information given in the problem to fill in the amount and percent columns of the table. Fill in the quantity column by multiplying the two expressions you wrote in each row. Use the expressions in the last column to write the equation.

20% solution 30% solution 22% solution

Amount of Solution, A

x 6

x6

Percent Concentration, r 0.20 0.30 0.22

Quantity of Substance, Q

0.20x 0.306 0.22x 6

2. Determine how the quantities of the substances in the solutions are related. Use the

fact that the sum of the quantities of the substances being mixed is equal to the quantity of the substance after mixing.

The sum of the quantities of the substances in the 20% solution and the 30% solution is equal to the quantity of the substance in the 22% solution. 24 gal of the 20% solution is required.

0.20x 0.306 0.22x 6 0.20x 1.80 0.22x 1.32 0.02x 1.80 1.32 0.02x 0.48 x 24

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Example 2

You Try It 2

A chemist wishes to make 2 L of an 8% acid solution by mixing a 10% acid solution and a 5% acid solution. How many liters of each solution should the chemist use?

A pharmacist dilutes 5 L of a 12% solution with a 6% solution. How many liters of the 6% solution are added to make an 8% solution?

Strategy

Your strategy

x L of 10% acid

+

(2 – x) L of 5% acid

=

2 L of 8% acid

• Liters of 10% solution: x Liters of 5% solution: 2 x Amount

Percent

x 2x 2

0.10 0.05 0.08


Quantity 0.10x 0.052 x 0.082

• The sum of the quantities before mixing is equal to the quantity after mixing.

Solution

Your solution

0.10x 0.052 x 0.082 0.10x 0.10 0.05x 0.16 0.05x 0.06 x 1.2 2 x 2 1.2 0.8 The chemist needs 1.2 L of the 10% solution and 0.8 L of the 5% solution.

Solution on p. S7


0.05x 0.10 0.16


Objective C

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SSM

Recall from Section 2.1 that an object traveling at a constant speed in a straight line is in uniform motion. The solution of a uniform motion problem is based on the equation rt d, where r is the rate of travel, t is the time spent traveling, and d is the distance traveled.

HOW TO A car leaves a town traveling at 40 mph. Two hours later, a second car leaves the same town, on the same road, traveling at 60 mph. In how many hours will the second car pass the first car?

Strategy for Solving a Uniform Motion Problem 1. For each object, write a numerical or variable expression for the rate, time, and

distance. The results can be recorded in a table.

The first car traveled 2 h longer than the second car. Unknown time for the second car: t Time for the first car: t 2

TA K E N O T E Use the information given in the problem to fill in the rate and time columns of the table. Find the expression in the distance column by multiplying the two expressions you wrote in each row.

First car Second car

Rate, r

Time, t

Distance, d

40 60

t2 t

40t 2 60t

First car

d = 40(t + 2)

Second car d = 60t


2. Determine how the distances traveled by the two objects are related. For example,

the total distance traveled by both objects may be known, or it may be known that the two objects traveled the same distance.

The two cars travel the same distance.

The second car will pass the first car in 4 h.

40t 2 60t 40t 80 60t 80 20t 4t

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Example 3

You Try It 3

Two cars, one traveling 10 mph faster than the other, start at the same time from the same point and travel in opposite directions. In 3 h they are 300 mi apart. Find the rate of each car.

Two trains, one traveling at twice the speed of the other, start at the same time on parallel tracks from stations that are 288 mi apart and travel toward each other. In 3 h, the trains pass each other. Find the rate of each train.

Strategy

Your strategy

• Rate of 1st car: r Rate of 2nd car: r 10

1st car 2nd car

Rate

Time

Distance

r r 10

3 3

3r 3r 10

• The total distance traveled by the two cars is 300 mi. Solution

Your solution

3r 3r 10 300 3r 3r 30 300 6r 30 300 6r 270 r 45 r 10 45 10 55 The first car is traveling 45 mph. The second car is traveling 55 mph. Example 4

You Try It 4

How far can the members of a bicycling club ride out into the country at a speed of 12 mph and return over the same road at 8 mph if they travel a total of 10 h?

A pilot flew out to a parcel of land and back in 5 h. The rate out was 150 mph, and the rate returning was 100 mph. How far away was the parcel of land?

Strategy

Your strategy

Out Back

Rate

Time

Distance

12 8

t 10 t

12t 810 t

• The distance out equals the distance back. Solution

Your solution

12t 810 t 12t 80 8t 20t 80 t 4 (The time is 4 h.) The distance out 12t 124 48 mi. The club can ride 48 mi into the country.

Solutions on p. S7


• Time spent riding out: t Time spent riding back: 10 t


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2.4 Exercises To solve value mixture problems

1.

An herbalist has 30 oz of herbs costing $2 per ounce. How many ounces of herbs costing $1 per ounce should be mixed with the 30 oz to produce a mixture costing $1.60 per ounce?

2.

The manager of a farmer’s market has 500 lb of grain that costs $1.20 per pound. How many pounds of meal costing $.80 per pound should be mixed with the 500 lb of grain to produce a mixture that costs $1.05 per pound?

3.

Find the cost per pound of a meatloaf mixture made from 3 lb of ground beef costing $1.99 per pound and 1 lb of ground turkey costing $1.39 per pound.

4.

Find the cost per ounce of a sunscreen made from 100 oz of a lotion that costs $2.50 per ounce and 50 oz of a lotion that costs $4.00 per ounce.

5.

A snack food is made by mixing 5 lb of popcorn that costs $.80 per pound with caramel that costs $2.40 per pound. How much caramel is needed to make a mixture that costs $1.40 per pound?

6.

A wild birdseed mix is made by combining 100 lb of millet seed costing $.60 per pound with sunflower seeds costing $1.10 per pound. How many pounds of sunflower seeds are needed to make a mixture that costs $.70 per pound?

7.

Ten cups of a restaurant’s house Italian dressing is made by blending olive oil costing $1.50 per cup with vinegar that costs $.25 per cup. How many cups of each are used if the cost of the blend is $.50 per cup?

8.

A high-protein diet supplement that costs $6.75 per pound is mixed with a vitamin supplement that costs $3.25 per pound. How many pounds of each should be used to make 5 lb of a mixture that costs $4.65 per pound?

9.

Find the cost per ounce of a mixture of 200 oz of a cologne that costs $5.50 per ounce and 500 oz of a cologne that costs $2.00 per ounce.

200 oz


Objective A

5 00 oz

10. Find the cost per pound of a trail mix made from 40 lb of raisins that cost $4.40 per pound and 100 lb of granola that costs $2.30 per pound.

120


11. A 20-ounce alloy of platinum that costs $220 per ounce is mixed with an alloy that costs $400 per ounce. How many ounces of the $400 alloy should be used to make an alloy that costs $300 per ounce?

12. How many liters of a blue dye that costs $1.60 per liter must be mixed with 18 L of anil that costs $2.50 per liter to make a mixture that costs $1.90 per liter?

13. The manager of a specialty food store combined almonds that cost $4.50 per pound with walnuts that cost $2.50 per pound. How many pounds of each were used to make a 100-pound mixture that costs $3.24 per pound?

14. A goldsmith combined an alloy that cost $4.30 per ounce with an alloy that cost $1.80 per ounce. How many ounces of each were used to make a mixture of 200 oz costing $2.50 per ounce?

15. Adult tickets for a play cost $6.00 and children’s tickets cost $2.50. For one performance, 370 tickets were sold. Receipts for the performance were $1723. Find the number of adult tickets sold.

$9.20 per pound

17. Find the cost per pound of sugar-coated breakfast cereal made from 40 lb of sugar that costs $1.00 per pound and 120 lb of corn flakes that cost $.60 per pound.

0 $5.5 per d poun

16. Tickets for a piano concert sold for $4.50 for each adult. Student tickets sold for $2.00 each. The total receipts for 1720 tickets were $5980. Find the number of adult tickets sold.

20 s nd pou

Objective B

To solve percent mixture problems

19. Forty ounces of a 30% gold alloy is mixed with 60 oz of a 20% gold alloy. Find the percent concentration of the resulting gold alloy.

20. One hundred ounces of juice that is 50% tomato juice is added to 200 oz of a vegetable juice that is 25% tomato juice. What is the percent concentration of tomato juice in the resulting mixture?


18. Find the cost per pound of a coffee mixture made from 8 lb of coffee that costs $9.20 per pound and 12 lb of coffee that costs $5.50 per pound.


121

21. How many gallons of a 15% acid solution must be mixed with 5 gal of a 20% acid solution to make a 16% acid solution?

22. How many pounds of a chicken feed that is 50% corn must be mixed with 400 lb of a feed that is 80% corn to make a chicken feed that is 75% corn?

23. A rug is made by weaving 20 lb of yarn that is 50% wool with a yarn that is 25% wool. How many pounds of the yarn that is 25% wool are used if the finished rug is 35% wool?

24. Five gallons of a light green latex paint that is 20% yellow paint is combined with a darker green latex paint that is 40% yellow paint. How many gallons of the darker green paint must be used to create a green paint that is 25% yellow paint?

25. How many gallons of a plant food that is 9% nitrogen must be combined with another plant food that is 25% nitrogen to make 10 gal of a solution that is 15% nitrogen?

26. A chemist wants to make 50 ml of a 16% acid solution by mixing a 13% acid solution and an 18% acid solution. How many milliliters of each solution should the chemist use? x ml of 13% acid

27. Five grams of sugar are added to a 45-gram serving of a breakfast cereal that is 10% sugar. What is the percent concentration of sugar in the resulting mixture?


28. A goldsmith mixes 8 oz of a 30% gold alloy with 12 oz of a 25% gold alloy. What is the percent concentration of the resulting alloy?

29. How many pounds of coffee that is 40% java beans must be mixed with 80 lb of coffee that is 30% java beans to make a coffee blend that is 32% java beans?

30. The manager of a garden shop mixes grass seed that is 60% rye grass with 70 lb of grass seed that is 80% rye grass to make a mixture that is 74% rye grass. How much of the 60% rye grass is used?

31. A hair dye is made by blending a 7% hydrogen peroxide solution and a 4% hydrogen peroxide solution. How many milliliters of each are used to make a 300-milliliter solution that is 5% hydrogen peroxide?

+

(50 – x) ml of 18% acid

=

50 ml of 16% acid

122


32. A tea that is 20% jasmine is blended with a tea that is 15% jasmine. How many pounds of each tea are used to make 5 lb of tea that is 18% jasmine?

33. How many ounces of pure chocolate must be added to 150 oz of chocolate topping that is 50% chocolate to make a topping that is 75% chocolate?

34. How many ounces of pure bran flakes must be added to 50 oz of cereal that is 40% bran flakes to produce a mixture that is 50% bran flakes?

35. Thirty ounces of pure silver is added to 50 oz of a silver alloy that is 20% silver. What is the percent concentration of the resulting alloy?

36. A clothing manufacturer has some pure silk thread and some thread that is 85% silk. How many kilograms of each must be woven together to make 75 kg of cloth that is 96% silk?

Objective C


37. Two small planes start from the same point and fly in opposite directions. The first plane is flying 25 mph slower than the second plane. In 2 h, the planes are 470 mi apart. Find the rate of each plane.

470 mi

38. Two cyclists start from the same point and ride in opposite directions. One cyclist rides twice as fast as the other. In 3 h, they are 81 mi apart. Find the rate of each cyclist.

40. A long-distance runner started on a course running at an average speed of 6 mph. One-half hour later, a second runner began the same course at an average speed of 7 mph. How long after the second runner starts will the second runner overtake the first runner?

41. A motorboat leaves a harbor and travels at an average speed of 9 mph toward a small island. Two hours later a cabin cruiser leaves the same harbor and travels at an average speed of 18 mph toward the same island. How many hours after the cabin cruiser leaves the harbor will it be alongside the motorboat?


39. Two planes leave an airport at 8 A.M., one flying north at 480 km h and the other flying south at 520 km h. At what time will they be 3000 km apart?


42. A 555-mile, 5-hour plane trip was flown at two speeds. For the first part of the trip, the average speed was 105 mph. For the remainder of the trip, the average speed was 115 mph. How long did the plane fly at each speed?

105 mph

115 mph

555 mi

43. An executive drove from home at an average speed of 30 mph to an airport where a helicopter was waiting. The executive boarded the helicopter and flew to the corporate offices at an average speed of 60 mph. The entire distance was 150 mi. The entire trip took 3 h. Find the distance from the airport to the corporate offices.

44. After a sailboat had been on the water for 3 h, a change in the wind direction reduced the average speed of the boat by 5 mph. The entire distance sailed was 57 mi. The total time spent sailing was 6 h. How far did the sailboat travel in the first 3 h?

45. A car and a bus set out at 3 P.M. from the same point headed in the same direction. The average speed of the car is twice the average speed of the bus. In 2 h the car is 68 mi ahead of the bus. Find the rate of the car.

46. A passenger train leaves a train depot 2 h after a freight train leaves the same depot. The freight train is traveling 20 mph slower than the passenger train. Find the rate of each train if the passenger train overtakes the freight train in 3 h.

100 mph

47. As part of flight training, a student pilot was required to fly to an airport and then return. The average speed on the way to the airport was 100 mph, and the average speed returning was 150 mph. Find the distance between the two airports if the total flying time was 5 h.


48. A ship traveling east at 25 mph is 10 mi from a harbor when another ship leaves the harbor traveling east at 35 mph. How long does it take the second ship to catch up to the first ship?

49. At 10 A.M., a plane leaves Boston, Massachusetts, for Seattle, Washington, a distance of 3000 mi. One hour later a plane leaves Seattle for Boston. Both planes are traveling at a speed of 500 mph. How many hours after the plane leaves Seattle will the planes pass each other?

50. At noon, a train leaves Washington, D.C., headed for Charleston, South Carolina, a distance of 500 mi. The train travels at a speed of 60 mph. At 1 P.M. a second train leaves Charleston headed for Washington, D.C., traveling at 50 mph. How long after the train leaves Charleston will the two trains pass each other?

150 mph

123

124


51. Two cyclists start at the same time from opposite ends of a course that is 51 mi long. One cyclist is riding at a rate of 16 mph, and the second cyclist is riding at a rate of 18 mph. How long after they begin will they meet?

51 mi

0 mi

52. A bus traveled on a level road for 2 h at an average speed that was 20 mph faster than its average speed on a winding road. The time spent on the winding road was 3 h. Find the average speed on the winding road if the total trip was 210 mi.

53. A bus traveling at a rate of 60 mph overtakes a car traveling at a rate of 45 mph. If the car had a 1-hour head start, how far from the starting point does the bus overtake the car?

54. A car traveling at 48 mph overtakes a cyclist who, riding at 12 mph, had a 3-hour head start. How far from the starting point does the car overtake the cyclist?

APPLYING THE CONCEPTS 55. Chemistry How many grams of pure water must be added to 50 g of pure acid to make a solution that is 40% acid?

56. Chemistry How many ounces of water must be evaporated from 50 oz of a 12% salt solution to produce a 15% salt solution?

58. Travel At 10 A.M., two campers left their campsite by canoe and paddled downstream at an average speed of 12 mph. They then turned around and paddled back upstream at an average rate of 4 mph. The total trip took 1 h. At what time did the campers turn around downstream?

59. Transportation A bicyclist rides for 2 h at a speed of 10 mph and then returns at a speed of 20 mph. Find the cyclist’s average speed for the trip.

60. Travel A car travels a 1-mile track at an average speed of 30 mph. At what average speed must the car travel the next mile so that the average speed for the 2 mi is 60 mph?


57. Automotive Technology A radiator contains 15 gal of a 20% antifreeze solution. How many gallons must be drained from the radiator and replaced by pure antifreeze so that the radiator will contain 15 gal of a 40% antifreeze solution?

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Section 2.5 / First-Degree Inequalities

2.5 Objective A

First-Degree Inequalities To solve an inequality in one variable

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The solution set of an inequality is a set of numbers, each element of which, when substituted for the variable, results in a true inequality. The inequality at the right is true if the variable is replaced by (for instance) 3, 1.98, or

2 . 3

x1 4 31 4 1.98 1 4 2 1 4 3

There are many values of the variable x that will make the inequality x 1 4 true. The solution set of the inequality is any number less than 5. The solution set can be written in set-builder notation as x x 5 .

Integrating

Technology See the appendix Keystroke Guide: Test for instructions on using a graphing calculator to graph the solution set of an inequality.

The graph of the solution set of x 1 4 is shown at the right.

−5 −4 −3 −2 −1 0

1

2

3

4

5

When solving an inequality, we use the Addition and Multiplication Properties of Inequalities to rewrite the inequality in the form variable constant or in the form variable constant. The Addition Property of Inequalities

If a b , then a c b c. If a b , then a c b c.


The Addition Property of Inequalities states that the same number can be added to each side of an inequality without changing the solution set of the inequality. This property is also true for an inequality that contains the symbol or . The Addition Property of Inequalities is used to remove a term from one side of an inequality by adding the additive inverse of that term to each side of the inequality. Because subtraction is defined in terms of addition, the same number can be subtracted from each side of an inequality without changing the solution set of the inequality.

HOW TO

Solve and graph the solution set: x 2 4

x24 x2242 x2

• Subtract 2 from each side of the inequality. • Simplify.

The solution set is x x 2 . −5 −4 −3 −2 −1

0

1

2

3

4

5

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Solve: 3x 4 2x 1

HOW TO

3x 4 2x 1 3x 4 2x 2x 1 2x

• Subtract 2x from each side of the inequality.

x 4 1 x 4 4 1 4

• Add 4 to each side of the inequality.

x 3

The solution set is x x 3 . The Multiplication Property of Inequalities is used to remove a coefficient from one side of an inequality by multiplying each side of the inequality by the reciprocal of the coefficient.

TA K E N O T E c 0 means c is a positive number. Note that the inequality symbols do not change.

c 0 means c is a

The Multiplication Property of Inequalities

Rule 1 If a b and c 0, then ac bc. If a b and c 0, then ac bc. Rule 2 If a b and c 0, then ac bc. If a b and c 0, then ac bc.

negative number. Note that the inequality symbols are reversed.

Here are some examples of this property. Rule 1 3 2 34 24 12 8

Rule 2

2 5 24 54 8 20

3 2 34 24 12 8

2 5 24 54 8 20

The Multiplication Property of Inequalities is also true for the symbols and . HOW TO TA K E N O T E Each side of the inequality is divided by a negative number; the inequality symbol must be reversed.

Solve: 3x 9

3x 9 3x 9 3 3 x 3

• Divide each side of the inequality by the coefficient 3. Because 3 is a negative number, the inequality symbol must be reversed.

The solution set is x x 3 .


Rule 1 states that when each side of an inequality is multiplied by a positive number, the inequality symbol remains the same. However, Rule 2 states that when each side of an inequality is multiplied by a negative number, the inequality symbol must be reversed. Because division is defined in terms of multiplication, when each side of an inequality is divided by a positive number, the inequality symbol remains the same. But when each side of an inequality is divided by a negative number, the inequality symbol must be reversed.


HOW TO

127

Solve: 3x 2 4

3x 2 4 3x 6 3x 6 3 3 x 2

• Subtract 2 from each side of the inequality. • Divide each side of the inequality by the coefficient 3.

The solution set is x x 2 . HOW TO

Solve: 2x 9 4x 5

2x 9 4x 5 2x 9 5 2x 14 2x 14 2 2 x 7

• Subtract 4x from each side of the inequality. • Add 9 to each side of the inequality. • Divide each side of the inequality by the coefficient 2. Reverse the inequality symbol.

The solution set is x x 7 . HOW TO

Solve: 5x 2 9x 32x 4

5x 2 9x 32x 4 5x 10 9x 6x 12 5x 10 3x 12 2x 10 12 2x 22 2x 22 2 2 x 11

• Use the Distributive Property to remove parentheses. • Subtract 3x from each side of the inequality. • Add 10 to each side of the inequality. • Divide each side of the inequality by the coefficient 2.


The solution set is x x 11 .

Example 1

You Try It 1

Solve and graph the solution set: x 3 4x 6

Solve and graph the solution set: 2x 1 6x 7

Solution

Your solution

x 3 4x 6 3x 3 6 3x 3 3x 3 3 3 x 1

• Subtract 4x from each side. • Subtract 3 from each side.

−5 −4 −3 −2 −1

0

1

2

3

4

5


The solution set is x x 1 . −5 − 4 −3 −2 −1

0

1

2

3

4

5

Solution on p. S7

128


Example 2

You Try It 2

Solve: 3x 5 3 23x 1

Solve: 5x 2 4 3x 2

Solution

Your solution

3x 5 3 23x 1 3x 5 3 6x 2 3x 5 1 6x 9x 5 1 9x 6 9x 6 9 9 2 x 3

x x 3 2

To solve a compound inequality

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A compound inequality is formed by joining two inequalities with a connective word such as and or or. The inequalities at the right are compound inequalities.

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2x 4 and 3x 2 8 2x 3 5 or x 2 5

The solution set of a compound inequality with the connective word and is the set of those elements that appear in the solution sets of both inequalities. Therefore, it is the intersection of the solution sets of the two inequalities. HOW TO

Solve: 2x 6 and 3x 2 4

2x 6 and x 3

x x 3

3x 2 4 3x 6 x 2

x x 2

• Solve each inequality.

The solution set of a compound inequality with and is the intersection of the solution sets of the two inequalities.

x x 3 x x 2 x 2 x 3 HOW TO

Solve: 3 2x 1 5

This inequality is equivalent to the compound inequality 3 2x 1 and 2x 1 5. 3 2x 1 and 2x 1 5 4 2x 2x 4 2 x x 2

x x 2

x x 2


The solution set of a compound inequality with and is the intersection of the solution sets of the two inequalities.

x x 2 x x 2 x 2 x 2


Objective B

Solution on p. S7


129

There is an alternative method for solving the inequality in the last example. HOW TO

Solve: 3 2x 1 5

3 2x 1 5 3 1 2x 1 1 5 1 4 2x 4 4 2x 4 2 2 2 2 x 2

• Subtract 1 from each of the three parts of the inequality.

• Divide each of the three parts of the inequality by the coefficient 2.

The solution set is x 2 x 2 . The solution set of a compound inequality with the connective word or is the union of the solution sets of the two inequalities. HOW TO

Solve: 2x 3 7 or 4x 1 3

2x 3 7 or 2x 4 x 2

x x 2

4x 1 3 4x 4 x 1

x x 1


Find the union of the solution sets.

x x 2 x x 1 x x 2 or x 1

Example 3

You Try It 3

Solve: 1 3x 5 4

Solve: 2 5x 3 13

Solution

Your solution


1 3x 5 4 1 5 3x 5 5 4 5 6 3x 9 6 3x 9 3 3 3 2 x 3

x 2 x 3

• Add 5 to each of the three parts.

• Divide each of the three parts by 3.

Example 4

You Try It 4

Solve: 11 2x 3 and 7 3x 4

Solve: 2 3x 11 or 5 2x 7

Solution

Your solution

11 2x 3 and 2x 14 x 7

x x 7

7 3x 4 3x 3 x 1

x x 1

x x 7 x x 1 x 1 x 7

Solutions on p. S7

130


Objective C


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Example 5

You Try It 5

A cellular phone company advertises two pricing plans. The first is $19.95 per month with 20 free minutes and $.39 per minute thereafter. The second is $23.95 per month with 20 free minutes and $.30 per minute thereafter. How many minutes can you talk per month for the first plan to cost less than the second?

The base of a triangle is 12 in. and the height is x 2 in. Express as an integer the maximum height of the triangle when the area is less than 50 in2.

Strategy

Your strategy

To find the number of minutes, write and solve an inequality using N to represent the number of minutes. Then N 20 is the number of minutes for which you are charged after the first free 20 min. Solution

Your solution

Cost of first plan cost of second plan 19.95 0.39N 20 23.95 0.30N 20 19.95 0.39N 7.8 23.95 0.30N 6 12.15 0.39N 17.95 0.30N 12.15 0.09N 17.95 0.09N 5.8 N 64.4

Example 6

You Try It 6

Find three consecutive positive odd integers whose sum is between 27 and 51.

An average score of 80 to 89 in a history course receives a B. Luisa Montez has grades of 72, 94, 83, and 70 on four exams. Find the range of scores on the fifth exam that will give Luisa a B for the course.

Strategy

Your strategy

To find the three integers, write and solve a compound inequality using n to represent the first odd integer. Solution

Your solution

Lower limit upper limit of the sum sum of the sum 27 n n 2 n 4 51 27 3n 6 51 27 6 3n 6 6 51 6 21 3n 45 21 3n 45 3 3 3 7 n 15 The three odd integers are 9, 11, and 13; or 11, 13, and 15; or 13, 15, and 17.

Solutions on p. S8


The first plan costs less if you talk less than 65 min.


131

2.5 Exercises Objective A 1.

To solve an inequality in one variable

State the Addition Property of Inequalities and give numerical examples of its use.

2.

3. Which numbers are solutions of the inequality x 7 3? a. 17 b. 8 c. 10 d. 0

State the Multiplication Property of Inequalities and give numerical examples of its use.

4. Which numbers are solutions of the inequality 2x 1 5? a. 6 b. 4 c. 3 d. 5

For Exercises 5 to 31, solve. For Exercises 5 to 10, graph the solution set. 5.

x3 2 −5 − 4 −3 −2 −1

6. 0

1

2

3

4

−5 −4 −3 −2 −1

5

7. 4x 8 −5 − 4 −3 −2 −1


0

1

2

3

4

5

0

1

2

3

4

5

0

1

2

3

4

5

8. 6x 12 0

1

2

3

4

−5 −4 −3 −2 −1

5

9. 2x 8 −5 − 4 −3 −2 −1

x42

10. 3x 9 0

1

2

3

4

−5 −4 −3 −2 −1

5

11.

3x 1 2x 2

12.

5x 2 4x 1

13.

2x 1 7

14.

3x 2 8

15.

5x 2 8

16.

4x 3 1

17.

6x 3 4x 1

18.

7x 4 2x 6

19.

8x 1 2x 13

20.

5x 4 2x 5

21.

4 3x 10

22.

2 5x 7

23.

7 2x 1

24.

3 5x 18

25.

3 4x 11

26.

2 x 7

27.

4x 2 x 11

28.

6x 5 x 10

29.

x 7 4x 8

30.

3x 1 7x 15

31.

3x 2 7x 4

132


For Exercises 32 to 47, solve. 32. 3x 5 2x 5

33.

3 3 x2 x 5 10

34.

5 1 x x4 6 6

2 3 7 1 x x 3 2 6 3

36.

7 3 2 5 x x 12 2 3 6

37.

1 3 7 x x2 2 4 4

35.

38. 6 2(x 4) 2x 10

39. 4(2x 1) 3x 2(3x 5)

40. 2(1 3x) 4 10 3(1 x)

41. 2 5(x 1) 3(x 1) 8

42. 2 2(7 2x) 3(3 x)

43. 3 2(x 5) x 5(x 1) 1

44.

10 13(2 x) 5(3x 2)

46. 3x 2(3x 5) 2 5(x 4)

Objective B

45. 3 4(x 2) 6 4(2x 1)

47. 12 2(3x 2) 5x 2(5 x)

To solve a compound inequality

Which set operation is used when a compound inequality is combined with or? b. Which set operation is used when a compound inequality is combined with and?

49.

Explain why writing 3 x 4 does not make sense.

For Exercises 50 to 83, solve. Write the solution set in set-builder notation. 50.

3x 6 and x 2 1

51.

x 3 1 and 2x 4

52.

x 2 5 or 3x 3

53. 2x 6 or x 4 1


48. a.



54.

2x 8 and 3x 6

55.

1 x 2 and 5x 10 2

56.

1 x 1 or 2x 0 3

57.

2 x 4 or 2x 8 3

58.

x 4 5 and 2x 6

59. 3x 9 and x 2 2

60.

5x 10 and x 1 6

61. 2x 3 1 and 3x 1 2

62.

7x 14 and 1 x 4

63. 4x 1 5 and 4x 7 1

64.

3x 7 10 or 2x 1 5

65. 6x 2 14 or 5x 1 11

66.

5 3x 4 16

67. 5 4x 3 21

68.

0 2x 6 4

69. 2 3x 7 1

70.

4x 1 11 or 4x 1 11

71. 3x 5 10 or 3x 5 10

72.

9x 2 7 and 3x 5 10

73. 8x 2 14 and 4x 2 10

74.

3x 11 4 or 4x 9 1

75. 5x 12 2 or 7x 1 13

133

134


76.

6 5x 14 24

77. 3 7x 14 31

78.

3 2x 7 and 5x 2 18

79. 1 3x 16 and 1 3x 16

80.

5 4x 21 or 7x 2 19

81. 6x 5 1 or 1 2x 7

82.

3 7x 31 and 5 4x 1

83. 9 x 7 and 9 2x 3

Objective C


84. Integers Five times the difference between a number and two is greater than the quotient of two times the number and three. Find the smallest integer that will satisfy the inequality.

85. Integers Two times the difference between a number and eight is less than or equal to five times the sum of the number and four. Find the smallest number that will satisfy the inequality.

87. Geometry The length of a rectangle is 5 cm less than twice the width. Express as an integer the maximum width of the rectangle when the perimeter is less than 60 cm.

88.

Telecommunications In 2003, the computer service America Online offered its customers a rate of $23.90 per month for unlimited use or $4.95 per month with 3 free hours plus $2.50 for each hour thereafter. Express as an integer the maximum number of hours you can use this service per month if the second plan is to cost you less than the first.

4w + 2 w


86. Geometry The length of a rectangle is 2 ft more than four times the width. Express as an integer the maximum width of the rectangle when the perimeter is less than 34 ft.


89. Telecommunications TopPage advertises local paging service for $6.95 per month for up to 400 pages, and $.10 per page thereafter. A competitor advertises service for $3.95 per month for up to 400 pages and $.15 per page thereafter. For what number of pages per month is the TopPage plan less expensive?

90. Consumerism Suppose PayRite Rental Cars rents compact cars for $32 per day with unlimited mileage, and Otto Rentals offers compact cars for $19.99 per day but charges $.19 for each mile beyond 100 mi driven per day. You want to rent a car for one week. How many miles can you drive during the week if Otto Rentals is to be less expensive than PayRite?

91. Consumerism During a weekday, to call a city 40 mi away from a certain pay phone costs $.70 for the first 3 minutes and $.15 for each additional minute. If you use a calling card, there is a $.35 fee and then the rates are $.196 for the first minute and $.126 for each additional minute. How long must a call be if it is to be cheaper to pay with coins rather than a calling card?

92. Temperature The temperature range for a week was between 14 F and 9 77 F. Find the temperature range in Celsius degrees. F C 32 5

93. Temperature The temperature range for a week in a mountain town was between 0 C and 30 C. Find the temperature range in Fahrenheit degrees. C

5(F 32) 9


94. Compensation You are a sales account executive earning $1200 per month plus 6% commission on the amount of sales. Your goal is to earn a minimum of $6000 per month. What amount of sales will enable you to earn $6000 or more per month?

95. Compensation George Stoia earns $1000 per month plus 5% commission on the amount of sales. George’s goal is to earn a minimum of $3200 per month. What amount of sales will enable George to earn $3200 or more per month?

96. Banking Heritage National Bank offers two different checking accounts. The first charges $3 per month and $.50 per check after the first 10 checks. The second account charges $8 per month with unlimited check writing. How many checks can be written per month if the first account is to be less expensive than the second account?

135

136


97. Banking Glendale Federal Bank offers a checking account to small businesses. The charge is $8 per month plus $.12 per check after the first 100 checks. A competitor is offering an account for $5 per month plus $.15 per check after the first 100 checks. If a business chooses the first account, how many checks does the business write monthly if it is assumed that the first account will cost less than the competitor’s account? 98. Education An average score of 90 or above in a history class receives an A grade. You have scores of 95, 89, and 81 on three exams. Find the range of scores on the fourth exam that will give you an A grade for the course. 99. Education An average of 70 to 79 in a mathematics class receives a C grade. A student has scores of 56, 91, 83, and 62 on four tests. Find the range of scores on the fifth test that will give the student a C for the course. 100. Integers and 78.

Find four consecutive integers whose sum is between 62

101. Integers Find three consecutive even integers whose sum is between 30 and 51.

APPLYING THE CONCEPTS 102. Let 2 x 3 and a 2x 1 b. a. Find the largest possible value of a. b. Find the smallest possible value of b. 103. Determine whether the following statements are always true, sometimes true, or never true. a. If a b, then a b. b. If a b and a 0, b 0, then

1 1 . a b

c. When dividing both sides of an inequality by an integer, we must reverse the inequality symbol. e. If a b 0 and c d 0, then ac bd. 104. The following is offered as the solution of 2 3(2x 4) 6x 5. 2 3(2x 4) 6x 5 2 6x 12 6x 5 6x 10 6x 5 6x 6x 10 6x 6x 5 10 5

• Use the Distributive Property. • Simplify. • Subtract 6x from each side.

Because 10 5 is a true inequality, the solution set is all real numbers. If this is correct, so state. If it is not correct, explain the incorrect step and supply the correct answer.


d. If a 1, then a2 a.

137

Section 2.6 / Absolute Value Equations and Inequalities

2.6 Objective A

Study

Tip

Before the class meeting in which your professor begins a new section, you should read each objective statement for that section. Next, browse through the objective material. The purpose of browsing through the material is to set the stage for your brain to accept and organize new information when it is presented to you. See AIM for Success, page xxvii.

Absolute Value Equations and Inequalities VIDEO & DVD

To solve an absolute value equation

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The absolute value of a number is its distance from zero on the number line. Distance is always a positive number or zero. Therefore, the absolute value of a number is always a positive number or zero. The distance from 0 to 3 or from 0 to 3 is 3 units.

3 3

3

3

−5 −4 −3 −2 −1 0

1

2

3

4

5

3 3

Absolute value can be used to represent the distance between any two points on the number line. The distance between two points on the number line is the absolute value of the difference between the coordinates of the two points.

The distance between point a and point b is given by b a . The distance between 4 and 3 on the number line is 7 units. Note that the order in which the coordinates are subtracted does not affect the distance.

7 −5 −4 −3 −2 −1 0

Distance 3 4 7 7

1

2

3

4

5

Distance 4 3 7 7

For any two numbers a and b, b a a b . An equation containing an absolute value symbol is called an absolute value equation. Here are three examples.

x 3

x 2 8

3x 4 5x 9

Solutions of an Absolute Value Equation Copyright © Houghton Mifflin Company. All rights reserved.

If a 0 and x a, then x a or x a.

For instance, given x 3, then x 3 or x 3 because 3 3 and 3 3. We can solve this equation as follows:

xx3 3

x 3

let x equal 3 and the opposite of 3.

Check:

x 3

3 3 33

• Remove the absolute value sign from x and

x 3

3 3 33

The solutions are 3 and 3.

138


x28 x6 Check:

Solve: x 2 8

HOW TO

x 2 x 8 2 8

• Remove the absolute value sign and rewrite as two equations.

x 10

x 2 8 6 2 8 88 88

• Solve each equation.

x 2 8 10 2 8 88 88


Solve: 5 3x 8 4

HOW TO

5 3x 8 4 5 3x 4

5 3x 4

• Solve for the absolute value. • Remove the absolute value sign and

5 3x 4

rewrite as two equations.

3x 1 1 x 3 Check:

3x 9

• Solve each equation.

x3

5 3x 8 4

5 3x 8 4 5 33 8 4 5 94 88 4 4

5 3 8 4 1 3

5 14 88 4 4

4 4

4 4

The solutions are

1 3

and 3.

You Try It 1

Solve: 2 x 12

Solve: 2x 3 5

Solution

Your solution

2 x 12 2 x 12 2 x 12 x 10 x 14 x 10 x 14

• Subtract 2. • Multiply by 1.


Example 2

You Try It 2

Solve: 2x 4

Solve: x 3 2

Solution

Your solution

2x 4

There is no solution to this equation because the absolute value of a number must be nonnegative.

Solutions on p. S8


Example 1

139


Example 3

You Try It 3

Solve: 3 2x 4 5

Solve: 5 3x 5 3

Solution

Your solution

3 2x 4 5 2x 4 8 • Subtract 3. 2x 4 8 • Multiply by 1. 2x 4 8 2x 4 8 2x 12 2x 4 x6 x 2 The solutions are 6 and 2.

Objective B

Solution on p. S8

To solve an absolute value inequality

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Recall that absolute value represents the distance between two points. For example, the solutions of the absolute value equation x 1 3 are the numbers whose distance from 1 is 3. Therefore, the solutions are 2 and 4. An absolute value inequality is an inequality that contains an absolute value symbol.

The solutions of the absolute value inequality x 1 3 are the numbers whose distance from 1 is less than 3. Therefore, the solutions are the numbers greater than 2 and less than 4. The solution set is x 2 x 4 .

Distance Distance less than 3 less than 3 −5 −4 −3 −2 −1 0

1

2

3

4

5

To solve an absolute value inequality of the form ax b c, solve the equivalent compound inequality c ax b c.


HOW TO

Solve: 3x 1 5

3x 1 5 5 3x 1 5 5 1 3x 1 1 5 1 4 3x 6 4 3x 6 3 3 3 4 x 2 3

The solution set is x

4 3

• Solve the equivalent compound inequality.

x 2 .

The solutions of the absolute value inequality x 1 2 are the numbers whose distance from 1 is greater than 2. Therefore, the solutions are the numbers that are less than 3 or greater than 1. The solution set of x 1 2 is

x x 3 or x 1 .

Distance greater than 2 −5 −4 −3 −2 −1

Distance greater than 2 0

1

2

3

4

5

140


TA K E N O T E Carefully observe the difference between the solution method of

ax b c shown here and that of ax b c shown on the preceding page.

To solve an absolute value inequality of the form ax b c, solve the equivalent compound inequality ax b c or ax b c. HOW TO

Solve: 3 2x 1

3 2x 1 or 2x 4 x 2

x x 2

3 2x 1 2x 2 x 1

x x 1


The solution of a compound inequality with or is the union of the solution sets of the two inequalities.

x x 2 x x 1 x x 2 or x 1 The rules for solving these absolute value inequalities are summarized below.

Solutions of Absolute Value Inequalities

To solve an absolute value inequality of the form ax b c, c 0, solve the equivalent compound inequality c ax b c. To solve an absolute value inequality of the form ax b c, solve the equivalent compound inequality ax b c or ax b c.

Example 4

Solve: 4x 3 5

You Try It 4

Solution

Solve the equivalent compound inequality.

Your solution

5 5 3 2 2 4

Solve: 3x 2 8

4x 3 5 4x 3 3 5 3 4x 8 4x 8 4 4

1 x 2 2 1 x x 2 2

Example 5

Solve: x 3 0

You Try It 5

Solution

The absolute value of a number is greater than or equal to zero, since it measures the number’s distance from zero on the number line. Therefore, the solution set of x 3 0 is the empty set.

Your solution

Solve: 3x 7 0

Solutions on p. S8



Example 6

Solve: x 4 2

You Try It 6

Solution

The absolute value of a number is greater than or equal to zero. Therefore, the solution set of

x 4 2 is the set of real numbers.

Your solution

Example 7

Solve: 2x 1 7

You Try It 7

Solution

Solve the equivalent compound inequality.

Your solution

141

Solve: 2x 7 1

Solve: 5x 3 8

2x 1 7 or 2x 1 7 2x 6 2x 8 x 3 x 4

x x 3

x x 4

x x 3 x x 4 x x 3 or x 4

Solutions on p. S8

Objective C


piston


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The tolerance of a component, or part, is the amount by which it is acceptable for the component to vary from a given measurement. For example, the diameter of a piston may vary from the given measurement of 9 cm by 0.001 cm. This is written 9 cm 0.001 cm and is read “9 centimeters plus or minus 0.001 centimeter.” The maximum diameter, or upper limit, of the piston is 9 cm 0.001 cm 9.001 cm. The minimum diameter, or lower limit, is 9 cm 0.001 cm 8.999 cm. The lower and upper limits of the diameter of the piston could also be found by solving the absolute value inequality d 9 0.001, where d is the diameter of the piston.

d 9 0.001

0.001 d 9 0.001 0.001 9 d 9 9 0.001 9 8.999 d 9.001 The lower and upper limits of the diameter of the piston are 8.999 cm and 9.001 cm.

142


Example 8

You Try It 8

The diameter of a piston for an automobile

A machinist must make a bushing that has a tolerance of 0.003 in. The diameter of the bushing is 2.55 in. Find the lower and upper limits of the diameter of the bushing.

is 3

5 16

in. with a tolerance of

1 64

in. Find the

lower and upper limits of the diameter of the piston.

Your strategy

Strategy

To find the lower and upper limits of the diameter of the piston, let d represent the diameter of the piston, T the tolerance, and L the lower and upper limits of the diameter. Solve the absolute value inequality L d T for L.

Solution

Ld 5 L3 16 1 64 1 5 3 64 16 19 3 64

Your solution

T 1 64

• d3

5 16

5 1 16 64 5 5 1 5 L3 3 3 16 16 64 16 21 L3 64 L3

The lower and upper limits of the diameter of the piston are 3

19 64

in. and 3

21 64

in.

Solution on p. S9



143


To solve an absolute value equation

Is 2 a solution of

x 8 6?

2. Is 2 a solution of

2x 5 9?

3.

Is 1 a solution of

3x 4 7?

4.

Is 1 a solution of

6x 1 5?


For Exercises 5 to 64, solve. 5.

x 7

8.

c 12

9.

y 6

10.

t 3

11.

a 7

12.

x 3

13.

x 4

14.

y 3

15.

t 3

16.

y 2

17.

x 2 3

18.

x 5 2

19.

y 5 3

20.

y 8 4

21.

a 2 0

22.

a 7 0

23.

x 2 4

24.

x 8 2

25.

3 4x 9

26.

2 5x 3

27.

2x 3 0

28.

5x 5 0

29.

3x 2 4

30.

2x 5 2

31.

x 2 2 3

32.

x 9 3 2

33.

3a 2 4 4

34.

2a 9 4 5

35.

2 y 3 4

36.

8 y 3 1

37.

2x 3 3 3

38.

4x 7 5 5

39.

2x 3 4 4

40.

3x 2 1 1

6.

a 2

7.

b 4

144


41.

6x 5 2 4

42.

4b 3 2 7

44.

5x 2 5 7

45.

3 x4 5

47.

8 2x 3 5

48.

50.

1 5a 2 3

53.

2x 8 12 2

56.

5 2x 1 8

59.

6 2x 4 3

62.

3 3 5x 2

43.

3t 2 3 4

46.

2 x5 4

8 3x 2 3

49.

2 3x 7 2

51.

8 3x 3 2

52.

6 5b 4 3

54.

3x 4 8 3

55.

2 3x 4 5

57.

5 2x 1 5

60.

8 3x 2 5

63.

5 2x 3

Objective B

58.

3 5x 3 3

61.

8 1 3x 1

64.

6 3 2x 2

To solve an absolute value inequality

65.

x 3

66.

x 5

67.

x 1 2

68.

x 2 1

69.

x 5 1

70.

x 4 3

71.

2 x 3

72.

3 x 2

73.

2x 1 5


For Exercises 65 to 94, solve.


74.

3x 2 4

75.

5x 2 12

76.

7x 1 13

77.

4x 3 2

78.

5x 1 4

79.

2x 7 5

80.

3x 1 4

81.

4 3x 5

82.

7 2x 9

83.

5 4x 13

84.

3 7x 17

85.

6 3x 0

86.

10 5x 0

87.

2 9x 20

88.

5x 1 16

89.

2x 3 2 8

90.

3x 5 1 7

91.

2 5x 4 2

92.

4 2x 9 3

93.

8 2x 5 3

94.

12 3x 4 7

Objective C


95. Mechanics The diameter of a bushing is 1.75 in. The bushing has a tolerance of 0.008 in. Find the lower and upper limits of the diameter of the bushing. Copyright © Houghton Mifflin Company. All rights reserved.

145

96. Mechanics A machinist must make a bushing that has a tolerance of 0.004 in. The diameter of the bushing is 3.48 in. Find the lower and upper limits of the diameter of the bushing.

97. Appliances An electric motor is designed to run on 220 volts plus or minus 25 volts. Find the lower and upper limits of voltage on which the motor will run.

1.75 in.

146


98. Computers A power strip is utilized on a computer to prevent the loss of programming by electrical surges. The power strip is designed to allow 110 volts plus or minus 16.5 volts. Find the lower and upper limits of voltage to the computer. 99. Automobiles erance of

1 32

A piston rod for an automobile is 9

5 8

in. long with a tol-

in. Find the lower and upper limits of the length of the

piston rod. 100. Automobiles erance of

1 64

A piston rod for an automobile is 9

3 8

in. long with a tol-

in. Find the lower and upper limits of the length of the

piston rod. Electronics The tolerance of the resistors used in electronics is given as a percent. Use your calculator for Exercises 101 to 104. 101.

Find the lower and upper limits of a 29,000-ohm resistor with a 2% tolerance.

102.


103.


104.

Find the lower and upper limits of a 56-ohm resistor with a 5% tolerance.

APPLYING THE CONCEPTS 105. For what values of the variable is the equation true? a. x 3 x 3 b. a 4 4 a

106. Write an absolute value inequality to represent all real numbers within 5 units of 2. 107. Replace the question mark with , , or . a. x y ? x y b. x y ? x y

c. x y ? x y e. xy ? xy

d.

x

x ? ,y0 y

y

108. Let x 2 and 3x 2 a. Find the smallest possible value of a.



147

Focus on Problem Solving Trial-and-Error Approach to Problem Solving

The questions below require an answer of always true, sometimes true, or never true. These problems are best solved by the trial-and-error method. The trialand-error method of arriving at a solution to a problem involves repeated tests or experiments. For example, consider the statement Both sides of an equation can be divided by the same number without changing the solution of the equation. The solution of the equation 6x 18 is 3. If we divide both sides of the equation by 2, the result is 3x 9 and the solution is still 3. So the answer “never true” has been eliminated. We still need to determine whether there is a case for which the statement is not true. Can we divide both sides of the equation by some number and get an equation for which the solution is not 3? If we divide both sides of the equation by 0, the result is

18 6x ; 0 0

the solution of this equation is not 3 because

the expressions on either side of the equals sign are undefined. Thus the statement is true for some numbers and not true for 0. The statement is sometimes true. For Exercises 1 to 10, determine whether the statement is always true, sometimes true, or never true. 1. Both sides of an equation can be multiplied by the same number without changing the solution of the equation. 2. For an equation of the form ax b, a 0, multiplying both sides of the equation by the reciprocal of a will result in an equation of the form x constant. 3. The Multiplication Property of Equations is used to remove a term from one side of an equation. 4. Adding 3 to each side of an equation yields the same result as subtracting 3 from each side of the equation.


5. An equation contains an equals sign. 6. The same variable term can be added to both sides of an equation without changing the solution of the equation. 7. An equation of the form ax b c cannot be solved if a is a negative number. 8. The solution of the equation

x 0

0 is 0.

9. An even integer is a multiple of 2. 10. In solving an equation of the form ax b cx d, subtracting cx from each side of the equation results in an equation with only one variable term in it.

148


Projects and Group Activities Water Displacement

When an object is placed in water, the object displaces an amount of water that is equal to the volume of the object. HOW TO A sphere with a diameter of 4 in. is placed in a rectangular tank of water that is 6 in. long and 5 in. wide. How much does the water level rise? Round to the nearest hundredth. V

4 r3 3

4 32 V 23 3 3

• Use the formula for the volume of a sphere. 1 2

1 2

• r d 4 2

Let x represent the amount of the rise in water level. The volume of the sphere will equal the volume displaced by the water. As shown at the left, this volume is the rectangular solid with width 5 in., length 6 in., and height x in. V LWH x

d = 4 in.

6 in.

• Use the formula for the volume of a rectangular solid.

32 65x 3

• Substitute

32 x 90

• The exact height that the water will fill is

1.12 x

• Use a calculator to find an approximation.

32 for V, 5 for W, and 6 for L. 3 32 16 . 90 45

The water will rise approximately 1.12 in. 20 cm

30 cm 16 in.

20 in. 12 in.

Figure 1

Figure 2

12 in.

Figure 3

1. A cylinder with a 2-centimeter radius and a height of 10 cm is submerged in a tank of water that is 20 cm wide and 30 cm long (see Figure 1). How much does the water level rise? Round to the nearest hundredth. 2. A sphere with a radius of 6 in. is placed in a rectangular tank of water that is 16 in. wide and 20 in. long (see Figure 2). The sphere displaces water until two-thirds of the sphere is submerged. How much does the water level rise? Round to the nearest hundredth. 3. A chemist wants to know the density of a statue that weighs 15 lb. The statue is placed in a rectangular tank of water that is 12 in. long and 12 in. wide (see Figure 3). The water level rises 0.42 in. Find the density of the statue. Round to the nearest hundredth. (Hint: Density weight volume)


5 in.

149

Chapter 2 Summary


Examples

An equation expresses the equality of two mathematical expressions. [2.1A, p. 73]

3 24x 5 x 4 is an equation.

A solution of an equation is a number that, when substituted for the variable, results in a true equation. [2.1A, p. 73]

2 is a solution of 2 3x 8 because 2 32 8 is a true equation.

To solve an equation means to find a solution of the equation. The goal is to rewrite the equation in the form variable constant, because the constant is the solution. [2.1B, p. 74]

The equation x 3 is in the form variable constant. The constant, 3, is the solution of the equation.

Equivalent equations are equations that have the same solution. [2.1B, p. 74]

x 3 7 and x 4 are equivalent equations because the solution of each equation is 4.

Multiplying each side of an equation that contains fractions by the LCM of the denominators is called clearing denominators. [2.2A, p. 92]

We can clear the denominators 2 3

from the equation x

1 2

3 4

by


multiplying each side of the equation by 12. Consecutive integers follow one another in order. [2.3A, p. 106]

5, 6, 7 are consecutive integers. 9, 8, 7 are consecutive integers.

The solution set of an inequality is a set of numbers, each element of which, when substituted in the inequality, results in a true inequality. [2.5A, p. 125]

Any number greater than 4 is a solution of the inequality x 4.

A compound inequality is formed by joining two inequalities with a connective word such as and or or. [2.5B, p. 128]

3x 6 and 2x 5 7 2x 1 3 or x 2 4

An absolute value equation is an equation that contains an absolute value symbol. [2.6A, p. 137]

x 2 3

An absolute value inequality is an inequality that contains an absolute value symbol. [2.6B, p. 139]

x 4 5

2x 3 6

The tolerance of a component or part is the amount by which it is acceptable for the component to vary from a given measurement. The maximum measurement is the upper limit. The minimum measurement is the lower limit. [2.6C, p. 141]

The diameter of a bushing is 1.5 in. with a tolerance of 0.005 in. The lower and upper limits of the diameter of the bushing are 1.5 in. 0.005 in.

150


Essential Rules and Procedures Addition Property of Equations [2.1B, p. 74] The same number can be added to each side of an equation without changing the solution of the equation.

If a b, then a c b c.

Examples x 5 3 x 5 5 3 5 x 8

Multiplication Property of Equations [2.1C, p. 75]

Each side of an equation can be multiplied by the same nonzero number without changing the solution of the equation. If a b and c 0, then ac bc.

2 x4 3

3 2

2 3 x 4 3 2 x6

Basic Percent Equation [2.1D, p. 77]

Percent Base Amount PBA

30% of what number is 24? PB A 0.30B 24 0.30B 24 0.30 0.30 B 80

Simple Interest Equation [2.1D, p. 78]

Interest Principle Rate Time I Prt

A credit card company charges an annual interest rate of 21% on the monthly unpaid balance on a card. Find the amount of interest charged on an unpaid balance of $232 for April. I Prt

I 2320.21

1 12

4.06

The interest charged is $4.06. Consecutive Integers [2.3A, p. 106]

n, n 1, n 2, . . .

The sum of three consecutive integers is 33.

Consecutive Even or Consecutive Odd Integers [2.3A, p. 106]

n, n 2, n 4, . . .

The sum of three consecutive odd integers is 33. n n 2 n 4 33

Value Mixture Equation [2.4A, p. 113] Amount Unit Cost Value AC V

A merchant combines coffee that costs $6 per pound with coffee that costs $3.20 per pound. How many pounds of each should be used to make 60 lb of a blend that costs $4.50 per pound? 6x 3.2060 x 4.5060


n n 1 n 2 33

Chapter 2 Summary

151

Percent Mixture Problems [2.4B, p. 115]

Amount of Percent of Quantity of solution concentration substance Ar Q

A silversmith mixed 120 oz of an 80% silver alloy with 240 oz of a 30% silver alloy. Find the percent concentration of the resulting silver alloy. 0.80120 0.30240 x360

Uniform Motion Equation [2.4C, p. 117]

Rate Time Distance rt d

Two planes are 1640 mi apart and are traveling toward each other. One plane is traveling 60 mph faster than the other plane. The planes meet in 2 h. Find the speed of each plane. 2r 2r 60 1640

Addition Property of Inequalities [2.5A, p. 125]

If a b, then a c b c. If a b, then a c b c.

x 3 2 x 3 3 2 3 x 5

Multiplication Property of Inequalities [2.5A, p. 126]

Rule 1 If a b and c 0, then ac bc. If a b and c 0, then ac bc.

3 x 12 4 4 4 3 x 12 3 4 3 x 16

Rule 2 If a b and c 0, then ac bc. If a b and c 0, then ac bc.

2x 8 8 2x

2 2 x 4

Solutions of an Absolute Value Equation [2.6A, p. 137]

If a 0 and x a, then x a or x a.

x 3 7 x37 x 10

x 3 7 x 4


Solutions of Absolute Value Inequalities [2.6B, p. 140]

To solve an absolute value inequality of the form ax b c, c 0, solve the equivalent compound inequality c ax b c.

x 5 9 9 x 5 9 9 5 x 5 5 9 5 4 x 14

To solve an absolute value inequality of the form ax b c, solve the equivalent compound inequality ax b c or ax b c.

x 5 9 x 5 9 or x 4 or

x5 9 x 14

152



Solve: 3t 3 2t 7t 15

2.

Solve: 3x 7 2

3.


4.

Solve: x 4 5

5.

Solve: 3x 4 and x 2 1

6.

Solve:

7.

2 4 Solve: x 3 9

8.

Solve: x 4 8 3

9.

Solve: 2x 5 3

11. Solve: 2a 3 54 3a

13.

Solve: 4x 5 3

15.

Solve:

17.

Solve: 3x 2 x 4 or 7x 5 3x 3

1 5 3 3 x x 2 8 4 2

10. Solve:

2x 3 2 3x 2 3 5

12. Solve: 5x 2 8 or 3x 2 4


16. Solve: 6 3x 3 2

18. Solve: 2x 3 2x 4 34 2x


3 x 3 2x 5 5


19. Solve: x 9 6

20. Solve:

2 3 x 3 4

21. Solve: 3x 21

22. Solve:

2 4 a 3 9

23. Solve: 3y 5 3 2y

24. Solve: 4x 5 x 6x 8

25. Solve: 3x 4 56 x

26. Solve:

27. Solve: 5x 8 3

28. Solve: 2x 9 8x 15


29. Solve:

2 5 3 x x1 3 8 4

2x 3 3x 2 1 4 2

30. Solve: 2 32x 4 4x 21 3x

31. Solve: 5 4x 1 7

32. Solve: 2x 3 8

33. Solve: 5x 8 0

34. Solve: 5x 4 2

35. Uniform Motion A ferry leaves a dock and travels to an island at an average speed of 16 mph. On the return trip, the ferry travels at an average speed of 12 mph. The total time for the trip is 2 the dock?

1 3

h. How far is the island from

153

154


36. Mixtures A grocer mixed apple juice that costs $4.20 per gallon with 40 gal of cranberry juice that costs $6.50 per gallon. How much apple juice was used to make cranapple juice costing $5.20 per gallon?

37. Compensation A sales executive earns $800 per month plus 4% commission on the amount of sales. The executive’s goal is to earn $3000 per month. What amount of sales will enable the executive to earn $3000 or more per month?

38. Integers Translate “four less than the product of five and a number is sixteen” into an equation and solve.

39. Mechanics The diameter of a bushing is 2.75 in. The bushing has a tolerance of 0.003 in. Find the lower and upper limits of the diameter of the bushing.

40. Integers The sum of two integers is twenty. Five times the smaller integer is two more than twice the larger integer. Find the two integers.

41. Education An average score of 80 to 90 in a psychology class receives a B grade. A student has scores of 92, 66, 72, and 88 on four tests. Find the range of scores on the fifth test that will give the student a B for the course.

43. Mixtures An alloy containing 30% tin is mixed with an alloy containing 70% tin. How many pounds of each were used to make 500 lb of an alloy containing 40% tin?

44. Automobiles ance of

1 32

3 8

A piston rod for an automobile is 10 in. long with a toler-

in. Find the lower and upper limits of the length of the piston rod.

d = 1680 mi


42. Uniform Motion Two planes are 1680 mi apart and are traveling toward each other. One plane is traveling 80 mph faster than the other plane. The planes meet in 1.75 h. Find the speed of each plane.

Chapter 2 Test


Chapter 2 Test 3 5 4 8

1.

Solve: x 2 4

2.

Solve: b

3.

3 5 Solve: y 4 8

4.

Solve: 3x 5 7

5.

Solve:

3 y26 4

6.

Solve: 2x 3 5x 8 2x 10

7.

Solve: 2a 2 3a 4 a 5

8.

Is 2 a solution of x2 3x 2x 6?

9.

Solve:

2x 1 3x 4 5x 9 3 6 9

10.

Solve: 3x 2 6x 7

Solve: 4x 1 5 or 2 3x 8

11.

What is 0.5% of 8?

12.

13.

Solve: 4 3x 7 and 2x 3 7

14. Solve: 3 5x 12

15. Solve: 2 2x 5 7

17. Solve: 4x 3 5

16. Solve: 3x 5 4

155

156


18. Consumerism Gambelli Agency rents cars for $12 per day plus 10¢ for every mile driven. McDougal Rental rents cars for $24 per day with unlimited mileage. How many miles a day can you drive a Gambelli Agency car if it is to cost you less than a McDougal Rental car?

19. Mechanics A machinist must make a bushing that has a tolerance of 0.002 in. The diameter of the bushing is 2.65 in. Find the lower and upper limits of the diameter of the bushing.

20. Integers The sum of two integers is fifteen. Eight times the smaller integer is one less than three times the larger integer. Find the integers.

21. Mixtures How many gallons of water must be mixed with 5 gal of a 20% salt solution to make a 16% salt solution?

22. Mixtures A butcher combines 100 lb of hamburger that costs $2.10 per pound with 60 lb of hamburger that costs $3.70 per pound. Find the cost of the hamburger mixture.

24. Uniform Motion Two trains are 250 mi apart and are traveling toward each other. One train is traveling 5 mph faster than the other train. The trains pass each other in 2 h. Find the speed of each train.

25. Mixtures How many ounces of pure water must be added to 60 oz of an 8% salt solution to make a 3% salt solution?


23. Uniform Motion A jogger runs a distance at a speed of 8 mph and returns the same distance running at a speed of 6 mph. Find the total distance that the jogger ran if the total time running was 1 hour and 45 minutes.


157


Cumulative Review Exercises 1.

Subtract: 6 (20) 8

3.

Subtract:

5.

Simplify:

7.

Simplify: 3x 8x (12x)

9.

Simplify: 16x

5 7 6 16

5 8

2

1 2

1 3 3 4

1 8

2.

Multiply: (2)(6)(4)

4.

Simplify: 42

6.

Evaluate 3(a c) 2ab when a 2, b 3, and c 4.

8.

Simplify: 2a (b) 7a 5b

3 2

3

10. Simplify: 4(9y)

11. Simplify: 2(x2 3x 2)

12. Simplify: 2(x 3) 2(4 x)

13. Simplify: 3[2x 4(x 3)] 2

14. Find A B given A {4, 2, 0, 2} and B {4, 0, 4, 8}.

15. Graph: {x x 3} {x x 2}

16. Is 3 a solution of x2 6x 9 x 3?

−5 −4 −3 −2 −1

17. Solve:

0

1

3 x 15 5

2

3

4

5

18. Solve: 7x 8 29

158


19. Solve: 13 9x 14

20. Solve: 5x 8 12x 13

21. Solve: 11 4x 2x 8

22. Solve: 8x 34x 5 2x 11

23. Solve: 3 22x 1 32x 2 1

24. Solve: 3x 2 5 and x 5 1

25. Solve: 3 2x 5

26. Solve: 3x 1 5

27.

28.

Write 55% as a fraction.


29. 25% of what number is 30?

30. Integers Translate “the sum of six times a number and thirteen is five less than the product of three and the number” into an equation and solve.

32. Mixtures How many grams of pure gold must be added to 100 g of a 20% gold alloy to make an alloy that is 36% gold?

33. Uniform Motion A sprinter ran to the end of a track at an average rate of 8 m/s and then jogged back to the starting point at an average rate of 3 m/s. The sprinter took 55 s to run to the end of the track and jog back. Find the length of the track.


31. Mixtures How many pounds of an oat flour that costs $.80 per pound must be mixed with 40 lb of a wheat flour that costs $.50 per pound to make a blend that costs $.60 per pound?

chapter

3

Geometry

OBJECTIVES

Section 3.1

A B C

To solve problems involving lines and angles To solve problems involving angles formed by intersecting lines To solve problems involving the angles of a triangle

Section 3.2

A


B This is an aerial view of the house of William Paca, who was a Maryland Patriot and a signer of the Declaration of Independence. The house’s large, formal garden has been restored to its original splendor. The best way to appreciate the shapes sculpted in the garden is to view it from above, like in this photo. Each geometric shape combines with the others to form the entire garden. Exercise 98 on page 193 shows you how to use a geometric formula first to determine the size of an area, and then to calculate how much grass seed is needed for an area of that size.

Need help? For online student resources, such as section quizzes, visit this textbook’s website at math.college.hmco.com/students.

To solve problems involving the perimeter of a geometric figure To solve problems involving the area of a geometric figure

Section 3.3

A B

To solve problems involving the volume of a solid To solve problems involving the surface area of a solid

PREP TEST Do these exercises to prepare for Chapter 3. 1.

Solve: x 47 90

2.

Solve: 32 97 x 180

3.

Simplify: 218 210

4.

Evaluate abc when a 2, b 3.14, and c 9.

5.

Evaluate xyz3 when x , y 3.14,

6.

Evaluate ab c when a 6,

and z 3.

1 2

b 25, and c 15.

GO FIGURE In a school election, one candidate for class president received more than 94%, but less than 100%, of the votes cast. What is the least possible number of votes cast?


4 3

Section 3.1 / Introduction to Geometry

3.1 Objective A

Study

Tip

Before you begin a new chapter, you should take some time to review previously learned skills. One way to do this is to complete the Prep Test. See page 160. This test focuses on the particular skills that will be required for the new chapter.

Introduction to Geometry To solve problems involving lines and angles

VIDEO & DVD

CD TUTOR

SSM

WEB

The word geometry comes from the Greek words for “earth” and “measure.” The original purpose of geometry was to measure land. Today geometry is used in many fields, such as physics, medicine, and geology, and is applied in such areas as mechanical drawing and astronomy. Geometric forms are also used in art and design. Three basic concepts of geometry are the point, line, and plane. A point is symbolized by drawing a dot. A line is determined by two distinct points and extends indefinitely in both directions, as the arrows on the line shown at the right indicate. This line ←→ contains points A and B and is represented by AB. A line can also be represented by a single letter, such as . A ray starts at a point and extends indefinitely in one direction. The point at which a ray starts is called the endpoint ofthe ray. The ray shown at → the right is denoted by AB. Point A is the endpoint of the ray. A line segment is part of a line and has two endpoints. The line segment shown at the right is denoted by AB. The distance between the endpoints of AC is denoted by AC. If B is a point on AC, then AC (the distance from A to C) is the sum of AB (the distance from A to B) and BC (the distance from B to C).


161

HOW TO

A

B

A

B

A

B

A

B AC = AB + BC

Given AB 22 cm and AC 31 cm, find BC.

AC AB BC

• Write an equation for the distances

31 22 BC

• Substitute the given distances for

between points on the line segment. AB and AC into the equation.

9 BC

• Solve for BC.

BC 9 cm Plane

In this section we will be discussing figures that lie in a plane. A plane is a flat surface and can be pictured as a table top or blackboard that extends in all directions. Figures that lie in a plane are called plane figures.

C

162

Chapter 3 / Geometry

Point of Interest Geometry is one of the oldest branches of mathematics. Around 350 B.C., the Greek mathematician Euclid wrote the Elements, which contained all of the known concepts of geometry. Euclid’s contribution was to unify various concepts into a single deductive system that was based on a set of axioms.

Lines in a plane can be intersecting or parallel. Intersecting lines cross at a point in the plane. Parallel lines never meet. The distance between them is always the same.

Intersecting Lines

Parallel Lines

The symbol means “is parallel to.” In the figure at the right, j k and AB CD. Note that j contains AB and k contains CD. Parallel lines contain parallel line segments.

A

B

C

D

An angle is formed by two rays with the same endpoint. The vertex of the angle is the point at which the two rays meet. The rays are called the sides of the angle.

j k

Side

Vertex Side

An angle can also be named by a variable written between the rays close to the vertex. In the figure at the right, x QRS and y SRT. Note that in this figure, more than two rays meet at R. In this case, the vertex cannot be used to name an angle.

B

An angle is measured in degrees. The symbol for degrees is a small raised circle, °. Probably because early Babylonians believed that Earth revolves around the sun in approximately 360 days, the angle formed by a circle has a measure of 360° (360 degrees).

r2

C

Q S x y R

T

Point of Interest The first woman mathematician for whom documented evidence exists is Hypatia (370–415). She lived in Alexandria, Egypt, and lectured at the Museum, the forerunner of our modern university. She made important contributions in mathematics, astronomy, and philosophy.

r1

A

360°


If A and C are points on rays r1 and r2, and B is the vertex, then the angle is called B or ABC, where is the symbol for angle. Note that either the angle is named by the vertex, or the vertex is the second point listed when the angle is named by giving three points. ABC could also be called CBA.


163

A protractor is used to measure an angle. Place the center of the protractor at the vertex of the angle with the edge of the protractor along a side of the angle. The angle shown in the figure below measures 58°.

70 110

80

90

100

100

90

80

110 70

12 0 60

13 0 50

58° 15

30

0

30 15 0

0 13

0 14 40

40 14 0

50

60 0 12

20 160

160 20

0

0

A 90° angle is called a right angle. The symbol J represents a right angle.

180

The corner of a page of this book is a good example of a 90° angle.

180

10 170

170 10

TA K E N O T E

90°


Perpendicular lines are intersecting lines that form right angles.

The symbol means “is perpendicular to.” In the figure at the right, p q and AB CD. Note that line p contains AB and line q contains CD. Perpendicular lines contain perpendicular line segments.

90°

90°

90°

90°

p A C

q D B

Complementary angles are two angles whose measures have the sum 90°. A B 70° 20° 90° A and B are complementary angles.

70° A

B

20°

164


Study

A 180° angle is called a straight angle.

Tip

A great many new vocabulary words are introduced in this chapter. All of these terms are in bold type. The bold type indicates that these are concepts you must know to learn the material. Be sure to study each new term as it is presented.

180°

AOB is a straight angle.

A

Supplementary angles are two angles whose measures have the sum 180°.

O

B

130° 50° B

A

A B 130° 50° 180° A and B are supplementary angles.

An acute angle is an angle whose measure is between 0° and 90°. B above is an acute angle. An obtuse angle is an angle whose measure is between 90° and 180°. A above is an obtuse angle.

Two angles that share a common side are adjacent angles. In the figure at the right, DAC and CAB are adjacent angles. DAC 45° and CAB 55°.

D

C 45°

DAB DAC CAB 45° 55° 100°

55° A

HOW TO In the figure at the right, EDG 80°. FDG is three times the measure of EDF. Find the measure of EDF.

E

D

Let x the measure of EDF. Then 3x the measure of FDG. Write an equation and solve for x, the measure of EDF.

TA K E N O T E Answers to application problems must have units, such as degrees, feet, dollars, or hours.

B

F

G

EDF FDG EDG x 3x 80 4x 80 x 20

Example 1

You Try It 1

Given MN 15 mm, NO 18 mm, and MP 48 mm, find OP.

Given QR 24 cm, ST 17 cm, and QT 62 cm, find RS.

M

N

O

Q

P

Solution

MN NO OP .MP 15 18 OP .48 33 OP .48 OP .15 OP 15 mm

R

S

T

Your solution

• MN 15, NO 18, MP 48

Solution on p. S9


EDF 20°


Example 2

You Try It 2

Given XY 9 m and YZ is twice XY, find XZ.

Given BC 16 ft and AB (BC), find AC.

X

Y

Z

XZ XZ XZ XZ XZ

XY YZ XY 2(XY) 9 2(9) 9 18 27

1 4

A

Solution

165

B

C

Your solution

• YZ is twice XY. • XY 9.

XZ 27 m

Example 3

You Try It 3

Find the complement of a 38° angle.

Find the supplement of a 129° angle.

Strategy

Your strategy

Complementary angles are two angles whose sum is 90°. To find the complement, let x represent the complement of a 38° angle. Write an equation and solve for x. Solution

Your solution

x 38° 90° x 52° The complement of a 38° angle is a 52° angle.

Example 4

You Try It 4

Find the measure of x.

Find the measure of a. a

x

118° 68°


47°

Strategy

Your strategy

To find the measure of x, write an equation using the fact that the sum of the measure of x and 47° is 90°. Solve for x. Solution

Your solution

x 47° 90° x 43° The measure of x is 43°. Solutions on p. S9


Objective B

Point of Interest Many cities in the New World, unlike those in Europe, were designed using rectangular street grids. Washington, D.C. was planned that way except that diagonal avenues were added, primarily for the purpose of enabling quick troop movement in the event the city required defense. As an added precaution, monuments of statuary were constructed at major intersections so that attackers would not have a straight shot down a boulevard.

To solve problems involving angles formed by intersecting lines

VIDEO & DVD

CD TUTOR

SSM

WEB

Four angles are formed by the intersection of two lines. If the two lines are perpendicular, then each of the four angles is a right angle.

If the two lines are not perpendicular, then two of the angles formed are acute angles and two of the angles are obtuse angles. The two acute angles are always opposite each other, and the two obtuse angles are always opposite each other. In the figure at the right, w and y are acute angles. x and z are obtuse angles. Two angles that are on opposite sides of the intersection of two lines are called vertical angles. Vertical angles have the same measure. w and y are vertical angles. x and z are vertical angles.

Two angles that share a common side are called adjacent angles. For the figure shown above, x and y are adjacent angles, as are y and z, z and w, and w and x. Adjacent angles of intersecting lines are supplementary angles.

x y

w

q

z

Vertical angles have the same measure. w y x z

Adjacent angles of intersecting lines are supplementary angles. x y y z z w w x

HOW TO Given that c 65°, find the measures of angles a, b, and d.

180° 180° 180° 180°

k b a

c d

a 65°

• a 5 c because

b c .180° b 65° .180° b .115°

• b is supplementary to c because

d 115°

• d 5 b because d and b are

a and c are vertical angles.

b and c are adjacent angles of intersecting lines.

vertical angles.

p


166

167


A line that intersects two other lines at different points is called a transversal.

t

Transversal

If the lines cut by a transversal t are parallel lines and the transversal is perpendicular to the parallel lines, all eight angles formed are right angles.

If the lines cut by a transversal t are parallel lines and the transversal is not perpendicular to the parallel lines, all four acute angles have the same measure and all four obtuse angles have the same measure. For the figure at the right,

1

2

t a d w z

b 1

c x

2

y


b d x z a c w y

Alternate interior angles are two nonadjacent angles that are on opposite sides of the transversal and lie between the parallel lines. In the figure above, c and w are alternate interior angles; d and x are alternate interior angles. Alternate interior angles have the same measure.

Alternate interior angles have the same measure.

Alternate exterior angles are two nonadjacent angles that are on opposite sides of the transversal and lie outside the parallel lines. In the figure above, a and y are alternate exterior angles; b and z are alternate exterior angles. Alternate exterior angles have the same measure.

Alternate exterior angles have the same measure.

Corresponding angles are two angles that are on the same side of the transversal and are both acute angles or are both obtuse angles. For the figure above, the following pairs of angles are corresponding angles: a and w, d and z, b and x, and c and y. Corresponding angles have the same measure.

Corresponding angles have the same measure.

c w d x

a y b z

a d b c

w z x y

168


t

HOW TO Given that 1 2 and c 58°, find the measures of f, h, and g.

b

a d

c f e

f c .58°

1

g 2

h

• c and f are alternate interior angles.

h c .58°

• c and h are corresponding angles.

g h .180° g 58° .180° g .122°

• g is supplementary to h.

Example 5

You Try It 5

Find x.

Find x. x + 30°

3x

x

x + 16º m

m

Strategy

Your strategy

The angles labeled are adjacent angles of intersecting lines and are, therefore, supplementary angles. To find x, write an equation and solve for x. Solution

Your solution

180° 180° 150° 75°

Example 6

Given 1 2, find x.

You Try It 6

Given 1 2, find x.

t y

t

x + 15º 1

2

y

2x

Strategy

1

3x x + 40º

2

Your strategy

2x y because alternate exterior angles have the same measure. (x 15°) y 180° because adjacent angles of intersecting lines are supplementary angles. Substitute 2x for y and solve for x. Solution

Your solution

(x 15°) 2x 180° 3x 15° 180° 3x 165° x 55° Solutions on p. S9


x (x 30°) 2x 30° 2x x

169


Objective C

To solve problems involving the angles of a triangle

VIDEO & DVD

CD TUTOR

If the lines cut by a transversal are not parallel lines, the three lines will intersect at three points. In the figure at the right, the transversal t intersects lines p and q. The three lines intersect at points A, B, and C. These three points define three line segments: AB, BC, and AC. The plane figure formed by these three line segments is called a triangle.

WEB

SSM

t

B

p C q

A

Each of the three points of intersection is the vertex of four angles. The angles within the region enclosed by the triangle are called interior angles. In the figure at the right, angles a, b, and c are interior angles. The sum of the measures of the interior angles of a triangle is 180°.

t

b

p

c

a

q ∠a + ∠b + ∠c = 180°

The Sum of the Measures of the Interior Angles of a Triangle

The sum of the measures of the interior angles of a triangle is 180°.

An angle adjacent to an interior angle is an exterior angle. In the figure at the right, angles m and n are exterior angles for angle a. The sum of the measures of an interior and an exterior angle is 180°.

m a n

∠a + ∠m = 180° ∠a + ∠n = 180°


Given that c 40° and HOW TO d 100°, find the measure of e. d and b are supplementary angles. d b 180° 100° b 180° b 80° The sum of the interior angles is 180°. c b a 40° 80° a 120° a a

180° 180° 180° 60°

a and e are vertical angles. e a 60°

t p c

b

d

a e

q

170


Example 7

You Try It 7

Given that y 55°, find the measures of angles a, b, and d.

Given that a 45° and x 100°, find the measures of angles b, c, and y.

m d

k

b

x b c

a

y m

a y k

Strategy

Your strategy

• To find the measure of angle a, use the fact that a and y are vertical angles. • To find the measure of angle b, use the fact that the sum of the measures of the interior angles of a triangle is 180°. • To find the measure of angle d, use the fact that the sum of an interior and an exterior angle is 180°. Your solution

Solution

a y 55° a b 90° 55° b 90° b 145° b

.180° .180° .180° .35°

Example 8

You Try It 8

Two angles of a triangle measure 53° and 78°. Find the measure of the third angle.

One angle in a triangle is a right angle, and one angle measures 34°. Find the measure of the third angle.

Strategy

Your strategy

To find the measure of the third angle, use the fact that the sum of the measures of the interior angles of a triangle is 180°. Write an equation using x to represent the measure of the third angle. Solve the equation for x. Solution

Your solution

x 53° 78° .180° x 131° .180° x .49° The measure of the third angle is 49°. Solutions on pp. S9–S10


d b .180° d 35° .180° d .145°



To solve problems involving lines and angles

Use a protractor to measure the angle. State whether the angle is acute, obtuse, or right. 1.

02.

4.

05.

0

3.

06.

Solve. 07. Find the complement of a 62° angle. 08. Find the complement of a 31° angle. 09. Find the supplement of a 162° angle. 10. Find the supplement of a 72° angle.


11. Given AB 12 cm, CD 9 cm, and AD 35 cm, find the length of BC.

A

12. Given AB 21 mm, BC 14 mm, and AD 54 mm, find the length of CD.

A

13. Given QR 7 ft and RS is three times the length of QR, find the length of QS.

Q

14. Given QR 15 in. and RS is twice the length of QR, find the length of QS.

Q

15. Given EF 20 m and FG is one-half the length of EF, find the length of EG.

E

B

B

C

C

R

D

D

S

R

S

F

G

171

172


16. Given EF 18 cm and FG is one-third the length of EF, find the length of EG.

E

F

17. Given LOM 53° and LON 139°, find the measure of MON.

G

M L N O

18. Given MON 38° and LON 85°, find the measure of LOM.

L M

O

N

Find the measure of x. 19.

20. 145° x

87° x

74°

24°

Given that LON is a right angle, find the measure of x. 21.

22. L

23. L M

4x x

x

x N

O

L

M

M

2x

24. L

O

N

x + 24°

x + 18° N

O

M

x O

N

25.

26. 127°

a a

53°

27.

28. a

76° 168°

a 172°

67°


Find the measure of a.


Find x. 29.

30. 4x

6x

3x

4x

2x

31.

32.

x + 36°

x + 20° 5x

2x

3x

2x

33.

4x

34. 3x

4x

2x

5x

x 2x

6x

3x

Solve. 35. Given a 51°, find the measure of b. b a

36. Given a 38°, find the measure of b.


b

Objective B

a

To solve problems involving angles formed by intersecting lines

Find the measure of x. 37.

p

38.

m

131°

x x

74° q

n

173

174


Find x. 39.

40.

j

m

3x + 22°

5x

4x + 36°

7x

k

n

Given that 1 2, find the measures of angles a and b. 41.

42.

t 38º

122º

1

a

t 1

a

2

b

2

b

43.

44.

t

t

1

47º

136º b

a

b

1

a

2

2

Given that 1 2, find x. 45.

46.

t 5x

t 1

3x

1

6x

4x

2

2

47.

48. x + 39º

3x 1 2

1

x + 20º 2

t

Objective C

t

To solve problems involving the angles of a triangle

Solve.

a

49. Given that a 95° and b 70°, find the measures of angles x and y.

b y

x


2x


50. Given that a 35° and b 55°, find the measures of angles x and y. b a x

y

51. Given that y 45°, find the measures of angles a and b. b

a y

52. Given that y 130°, find the measures of angles a and b. y

b

53. Given that AO OB, express in terms of x the number of degrees in BOC.

a

B

A x


O

54. Given that AO OB, express in terms of x the number of degrees in AOC.

C

B

A

x + 15º C

55. One angle in a triangle is a right angle, and one angle is equal to 30°. What is the measure of the third angle?

56. A triangle has a 45° angle and a right angle. Find the measure of the third angle.

O

175

176


57. Two angles of a triangle measure 42° and 103°. Find the measure of the third angle. 58. Two angles of a triangle measure 62° and 45°. Find the measure of the third angle. 59. A triangle has a 13° angle and a 65° angle. What is the measure of the third angle? 60. A triangle has a 105° angle and a 32° angle. What is the measure of the third angle?

APPLYING THE CONCEPTS 61. a. What is the smallest possible whole number of degrees in an angle of a triangle? b. What is the largest possible whole number of degrees in an angle of a triangle? 62. Cut out a triangle and then tear off two of the angles, as shown at the right. Position the pieces you tore off so that angle a is adjacent to angle b and angle c is adjacent to angle b (on the other side). Describe what you observe. What does this demonstrate?

b

a

c

63. Construct a triangle with the given angle measures. a. 45°, 45°, and 90° b. 30°, 60°, and 90° c. 40°, 40°, and 100° 64. Determine whether the statement is always true, sometimes true, or never true. a. Two lines that are parallel to a third line are parallel to each other. b. A triangle contains two acute angles. c. Vertical angles are complementary angles.

y

65. For the figure at the right, find the sum of the measures of angles x, y, and z. For the figure at the right, explain why a b x. Write a rule that describes the relationship between an exterior angle of a triangle and the opposite interior angles. Use the rule to write an equation involving angles a, c, and z.

67.

If AB and CD intersect at point O, and AOC BOC, explain why AB CD.

68.

Do some research on the principle of reflection. Explain how this principle applies to the operation of a periscope and to the game of billiards.

z b

x Copyright © Houghton Mifflin Company. All rights reserved.

66.

a c

Section 3.2 / Plane Geometric Figures

3.2 Objective A

177

Plane Geometric Figures To solve problems involving the perimeter of a geometric figure

VIDEO & DVD

CD TUTOR

SSM

WEB

A polygon is a closed figure determined by three or more line segments that lie in a plane. The line segments that form the polygon are called its sides. The figures below are examples of polygons.

A

Point of Interest Although a polygon is defined in terms of its sides (see the definition above), the word actually comes from the Latin word polygonum, which means “having many angles.” This is certainly the case for a polygon.

B

C

D

A regular polygon is one in which each side has the same length and each angle has the same measure. The polygons in Figures A, C, and D above are regular polygons. The name of a polygon is based on the number of its sides. The table below lists the names of polygons that have from 3 to 10 sides. Number of Sides

Name of the Polygon

3 4 5 6 7 8 9 10


The Pentagon in Arlington, Virginia

E

Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon

Triangles and quadrilaterals are two of the most common types of polygons. Triangles are distinguished by the number of equal sides and also by the measures of their angles. C C

A An isosceles triangle has two sides of equal length. The angles opposite the equal sides are of equal measure. AC = BC ∠A = ∠B

C

B

A

B

The three sides of an equilateral triangle are of equal length. The three angles are of equal measure. AB = BC = AC ∠A = ∠B = ∠C

A A scalene triangle has no two sides of equal length. No two angles are of equal measure.

B


C

A

TA K E N O T E The diagram below shows the relationships among all quadrilaterals. The description of each quadrilateral is given within an example of that quadrilateral.

A

C

B

B

A

An acute triangle has three acute angles.

B

C

An obtuse triangle has one obtuse angle.

A right triangle has a right angle.

Quadrilaterals are also distinguished by their sides and angles, as shown below. Note that a rectangle, a square, and a rhombus are different forms of a parallelogram.

Rectangle

Parallelogram

Square

Opposite sides parallel Opposite sides equal in length All angles measure 90° Diagonals equal in length

Opposite sides parallel Opposite sides equal in length Opposite angles equal in measure

Quadrilateral

Rhombus

Four-sided polygon

Trapezoid

Opposite sides parallel All sides equal in length Opposite angles equal in measure

Opposite sides parallel All sides equal in length All angles measure 90° Diagonals equal in length

Isosceles Trapezoid Two sides parallel

Two sides parallel

Nonparallel sides equal in length

The perimeter of a plane geometric figure is a measure of the distance around the figure. Perimeter is used in buying fencing for a lawn or determining how much baseboard is needed for a room. The perimeter of a triangle is the sum of the lengths of the three sides.

Perimeter of a Triangle

a

b

Let a, b, and c be the lengths of the sides of a triangle. The perimeter, P, of the triangle is given by P a b c. c P=a+b+c

HOW TO Find the perimeter of the triangle shown at the right. P 5 7 10 22 The perimeter is 22 ft.

7 ft 5 ft 10 ft


178

179


The perimeter of a quadrilateral is the sum of the lengths of its four sides. Point of Interest Leonardo DaVinci painted the Mona Lisa on a rectangular canvas whose height was approximately 1.6 times its width. Rectangles with these proportions, called golden rectangles, were used extensively in Renaissance art.

A rectangle is a quadrilateral with opposite sides of equal length. Usually the length, L, of a rectangle refers to the length of one of the longer sides of the rectangle, and the width, W, refers to the length of one of the shorter sides. The perimeter can then be represented by P L W L W.

L W

W

The formula for the perimeter of a rectangle is derived by combining like terms.

L P=L+W+L+W

P 2L 2W

Perimeter of a Rectangle

Let L represent the length and W the width of a rectangle. The perimeter, P, of the rectangle is given by P 2L 2W.

HOW TO the right.

Find the perimeter of the rectangle shown at

P 2L 2W

5m

P 2(5) 2(2)

• The length is 5 m. Substitute 5 for L.

P 10 4

• Solve for P.

The width is 2 m. Substitute 2 for W. 2m

P 14 The perimeter is 14 m.


A square is a rectangle in which each side has the same length. Let s represent the length of each side of a square. Then the perimeter of the square can be represented by P s s s s. The formula for the perimeter of a square is derived by combining like terms.

s s

s

s P=s+s+s+s

P 4s

Perimeter of a Square

Let s represent the length of a side of a square. The perimeter, P, of the square is given by P 4s.

HOW TO the right.

Find the perimeter of the square shown at

P 4s 4(8) 32 The perimeter is 32 in.

8 in.

180


A circle is a plane figure in which all points are the same distance from point O, which is called the center of the circle. The diameter of a circle is a line segment across the circle through point O. AB is a diameter of the circle at the right. The variable d is used to designate the diameter of a circle.

C

A

B

O

The radius of a circle is a line segment from the center of the circle to a point on the circle. OC is a radius of the circle at the right. The variable r is used to designate a radius of a circle.

Archimedes (c. 287–212 B.C.) was the mathematician who gave us the approximate 22 1 value of as 3 . He 7 7 actually showed that was 10 1 between 3 and 3 . The 71 7 10 approximation 3 is closer to 71 the exact value of , but it is more difficult to use.

d 2r or r

The distance around a circle is called the circumference. The circumference, C, of a circle is equal to the product of (pi) and the diameter.

C d

Because d 2r, the formula for the circumference can be written in terms of r.

C 2 r

1 d 2

The Circumference of a Circle

The circumference, C, of a circle with diameter d and radius r is given by C d or C 2 r.

The formula for circumference uses the number , which is an irrational number. The value of can be approximated by a fraction or by a decimal.

22 or 3.14 7

The key on a scientific calculator gives a closer approximation of than 3.14. Use a scientific calculator to find approximate values in calculations involving .

HOW TO C d

Integrating

Technology The key on your calculator can be used to find decimal approximations for expressions that contain . To perform the calculation at the right, enter 6 .

C (6) C 6

Find the circumference of a circle with a diameter of 6 in. • The diameter of the circle is given. Use the circumference formula that involves the diameter. d 5 6.

• The exact circumference of the circle is 6 p in.

C 18.85

• An approximate measure is found by using the p key on a calculator.

The circumference is approximately 18.85 in.


Point of Interest

The length of the diameter of a circle is twice the length of the radius.


Example 1

You Try It 1

A carpenter is designing a square patio with a perimeter of 44 ft. What is the length of each side?

The infield of a softball field is a square with each side of length 60 ft. Find the perimeter of the infield.

Strategy

Your strategy

181

To find the length of each side, use the formula for the perimeter of a square. Substitute 44 for P and solve for s. Solution

Your solution

P 4s 44 4s 11 s The length of each side of the patio is 11 ft. Example 2

You Try It 2

The dimensions of a triangular sail are 18 ft, 11 ft, and 15 ft. What is the perimeter of the sail?

What is the perimeter of a standard piece of

Strategy

Your strategy

1 2

typing paper that measures 8 in. by 11 in.?

To find the perimeter, use the formula for the perimeter of a triangle. Substitute 18 for a, 11 for b, and 15 for c. Solve for P. Solution

Your solution

Pabc P 18 11 15 P 44


The perimeter of the sail is 44 ft. Example 3

You Try It 3

Find the circumference of a circle with a radius of 15 cm. Round to the nearest hundredth.

Find the circumference of a circle with a diameter of 9 in. Give the exact measure.

Strategy

Your strategy

To find the circumference, use the circumference formula that involves the radius. An approximation is asked for; use the key on a calculator. r 15. Solution

Your solution

C 2 r 2 (15) 30 94.25 The circumference is approximately 94.25 cm. Solutions on p. S10

182


To solve problems involving the area of a geometric figure

Objective B

VIDEO & DVD

CD TUTOR

WEB

SSM

Area is the amount of surface in a region. Area can be used to describe the size of a rug, a parking lot, a farm, or a national park. Area is measured in square units.

Polygonal numbers are whole numbers that can be represented as regular geometric figures. For example, a square number is one that can be represented as a square array. o

o o o o

o o o o o o o o o

1

4

9

o o o o o o o o 16

o o o o

o o o o

The square numbers are 1, 4, 9, 16, 25, . . . . They can be represented as 12, 22, 32, 42, 52, . . . .

A square that measures 1 in. on each side has an area of 1 square inch, written 1 in2. A square that measures 1 cm on each side has an area of 1 square centimeter, written 1 cm2.

1 in2 1 cm2

Larger areas can be measured in square feet (ft2), square meters (m2), square miles (mi2), acres (43,560 ft2), or any other square unit. The area of a geometric figure is the number of squares that are necessary to cover the figure. In the figures below, two rectangles have been drawn and covered with squares. In the figure on the left, 12 squares, each of area 1 cm2, were used to cover the rectangle. The area of the rectangle is 12 cm2. In the figure on the right, 6 squares, each of area 1 in2, were used to cover the rectangle. The area of the rectangle is 6 in2.

The area of the rectangle is 12 cm2. The area of the rectangle is 6 in2.

Note from the above figures that the area of a rectangle can be found by multiplying the length of the rectangle by its width.

Area of a Rectangle

Let L represent the length and W the width of a rectangle. The area, A, of the rectangle is given by A LW.

HOW TO Find the area of the rectangle shown at the right. 7m

A LW 11(7) 77 The area is 77 m2.

11 m


Point of Interest

183


A square is a rectangle in which all sides are the same length. Therefore, both the length and the width of a square can be represented by s, and A LW s s s2. Area of a Square

s

Let s represent the length of a side of a square. The area, A, of the square is given by A s 2. A = s ⋅ s = s2

HOW TO Find the area of the square shown at the right. 9 mi

A s2 92 81 The area is 81 mi2.

Figure ABCD is a parallelogram. BC is the base, b, of the parallelogram. AE, perpendicular to the base, is the height, h, of the parallelogram.

A

D

h B

E

C b

Any side of a parallelogram can be designated as the base. The corresponding height is found by drawing a line segment perpendicular to the base from the opposite side. A rectangle can be formed from a parallelogram by cutting a right triangle from one end of the parallelogram and attaching it to the other end. The area of the resulting rectangle will equal the area of the original parallelogram.

h=W

h


b

b=L

Area of a Parallelogram

Let b represent the length of the base and h the height of a parallelogram. The area, A, of the parallelogram is given by A bh.

Find the area of the paralleloHOW TO gram shown at the right. A bh 12 6 72 The area is 72 m2.

6m 12 m

184


Figure ABC is a triangle. AB is the base, b, of the triangle. CD, perpendicular to the base, is the height, h, of the triangle.

C

h A

Any side of a triangle can be designated as the base. The corresponding height is found by drawing a line segment perpendicular to the base from the vertex opposite the base.

1 2

B b

h b

Consider the triangle with base b and height h shown at the right. By extending a line from C parallel to the base AB and equal in length to the base, and extending a line from B parallel to AC and equal in length to AC, a parallelogram is formed. The area of the parallelogram is bh and is twice the area of the triangle. Therefore, the area of the triangle is one-half the area of the parallelogram, or

D

C

h A

B

b

bh.

Area of a Triangle

Let b represent the length of the base and h the height of a triangle. The area, A, of the 1 2

triangle is given by A bh.

Integrating HOW TO Find the area of a triangle with a base of 18 cm and a height of 6 cm. A

6 cm

1 1 bh 18 6 54 2 2

18 cm

The area is 54 cm2. Figure ABCD is a trapezoid. AB is one base, b1, of the trapezoid, and CD is the other base, b2. AE, perpendicular to the two bases, is the height, h. In the trapezoid at the right, the line segment BD divides the trapezoid into two triangles, ABD and BCD. In triangle ABD, b1 is the base and h is the height. In triangle BCD, b2 is the base and h is the height. The area of the trapezoid is the sum of the areas of the two triangles.

A

D

B

C

E A

b1

B

h D

b2

Area of trapezoid ABCD area of triangle ABD area of triangle BCD

1 1 1 b1 h b2 h h(b1 b2) 2 2 2

C


Technology To calculate the area of the triangle shown at the right, you can enter 18 6 2 or .5 18 6 .

185


Area of a Trapezoid

Let b1 and b2 represent the lengths of the bases and h the height of a trapezoid. The area, 1 2

A, of the trapezoid is given by A hb1 b2.

HOW TO Find the area of a trapezoid that has bases measuring 15 in. and 5 in. and a height of 8 in. A

15 in. 8 in.

1 h(b1 b2) 2

5 in.

1 8(15 5) 4(20) 80 2

The area is 80 in2.

The area of a circle is equal to the product of and the square of the radius.

r

A = pr2

Area of a Circle


The area, A, of a circle with radius r is given by A r 2.

Integrating

Technology To approximate 36 on your calculator, enter 36 .

HOW TO A r2 A (6)2 A (36)

Find the area of a circle that has a radius of 6 cm. • Use the formula for the area of a

6 cm

circle. r 5 6

A 36

• The exact area of the circle is 36 p cm2.

A 113.10

• An approximate measure is found by using the p key on a calculator.

The approximate area of the circle is 113.10 cm2.

For your reference, all of the formulas for the perimeters and areas of the geometric figures presented in this section are listed in the Chapter Summary located at the end of this chapter.

186


Example 4

You Try It 4

The Parks and Recreation Department of a city plans to plant grass seed in a playground that has the shape of a trapezoid, as shown below. Each bag of grass seed will seed 1500 ft2. How many bags of grass seed should the department purchase?

An interior designer decides to wallpaper two walls of a room. Each roll of wallpaper will cover 30 ft2. Each wall measures 8 ft by 12 ft. How many rolls of wallpaper should be purchased?

80 ft 64 ft 115 ft

Strategy

Your strategy

To find the number of bags to be purchased: • Use the formula for the area of a trapezoid to find the area of the playground. • Divide the area of the playground by the area one bag will seed (1500). Solution

Your solution

1 A h(b1 b2 ) 2 1 A 64(80 115) 2 A 6240 • The area of the playground is 6240 ft2.

6240 1500 4.16

Example 5

You Try It 5

Find the area of a circle with a diameter of 5 ft. Give the exact measure.

Find the area of a circle with a radius of 11 cm. Round to the nearest hundredth.

Strategy

Your strategy

To find the area: • Find the radius of the circle. • Use the formula for the area of a circle. Leave the answer in terms of . Solution

Your solution

1 1 r d (5) 2.5 2 2 A r2 (2.5)2 (6.25) 6.25 The area of the circle is 6.25 ft2. Solutions on p. S10


Because a portion of a fifth bag is needed, 5 bags of grass seed should be purchased.


187


To solve problems involving the perimeter of a geometric figure

Name each polygon. 1.

2.

3.

4.

7.

8.

11.

12.

Classify the triangle as isosceles, equilateral, or scalene. 5.

6.

Classify the triangle as acute, obtuse, or right. 9.

10.

Find the perimeter of the figure. 13.

14.

15.

20 in.

12 in.


7 cm

3.5 ft

24 in. 11 cm

16.

17.

9m

18. 1 2

2 in.

8m

12 m

3.5 ft

13 mi 1 2

2 in.

10 m 10.5 mi

188


Find the circumference of the figure. Give both the exact value and an approximation to the nearest hundredth. 19.

4 cm

22.

20.

21. 12 m

5.5 mi

23. 18 in.

24. 17 ft 6.6 km

Solve. 25. The lengths of the three sides of a triangle are 3.8 cm, 5.2 cm, and 8.4 cm. Find the perimeter of the triangle. 26. The lengths of the three sides of a triangle are 7.5 m, 6.1 m, and 4.9 m. Find the perimeter of the triangle. 27. The length of each of two sides of an isosceles triangle is 2

1 2

cm. The

third side measures 3 cm. Find the perimeter of the triangle. 28. The length of each side of an equilateral triangle is 4

1 2

C

in. Find the

perimeter of the triangle. A

B

1 2

30. Find the perimeter of a rectangle that has a length of 5 ft and a width of 4 ft. 31. The length of each side of a square is 12.2 cm. Find the perimeter of the square. 32. Find the perimeter of a square that measures 0.5 m on each side. 33. Find the perimeter of a regular pentagon that measures 3.5 in. on each side.


29. A rectangle has a length of 8.5 m and a width of 3.5 m. Find the perimeter of the rectangle.


189

34. What is the perimeter of a regular hexagon that measures 8.5 cm on each side? 35. The radius of a circle is 4.2 cm. Find the length of a diameter of the circle. 36. The diameter of a circle is 0.56 m. Find the length of a radius of the circle. 37. Find the circumference of a circle that has a diameter of 1.5 in. Give the exact value. 38. The diameter of a circle is 4.2 ft. Find the circumference of the circle. Round to the nearest hundredth. 39. The radius of a circle is 36 cm. Find the circumference of the circle. Round to the nearest hundredth. 40. Find the circumference of a circle that has a radius of 2.5 m. Give the exact value. 41. Fencing How many feet of fencing should be purchased for a rectangular garden that is 18 ft long and 12 ft wide? 42. Quilting How many meters of binding are required to bind the edge of a rectangular quilt that measures 3.5 m by 8.5 m? 43. Carpeting Wall-to-wall carpeting is installed in a room that is 12 ft long and 10 ft wide. The edges of the carpet are nailed to the floor. Along how many feet must the carpet be nailed down? 44. Fencing The length of a rectangular park is 55 yd. The width is 47 yd. How many yards of fencing are needed to surround the park? L


45. Playgrounds The perimeter of a rectangular playground is 440 ft. If the width is 100 ft, what is the length of the playground?

100 ft

46. Gardens A rectangular vegetable garden has a perimeter of 64 ft. The length of the garden is 20 ft. What is the width of the garden? 47. Banners Each of two sides of a triangular banner measures 18 in. If the perimeter of the banner is 46 in., what is the length of the third side of the banner? 48. The perimeter of an equilateral triangle is 13.2 cm. What is the length of each side of the triangle?

s

s

s

190


49. Framing The perimeter of a square picture frame is 48 in. Find the length of each side of the frame. 50. Carpeting A square rug has a perimeter of 32 ft. Find the length of each edge of the rug. 51. The circumference of a circle is 8 cm. Find the length of a diameter of the circle. Round to the nearest hundredth. 52. The circumference of a circle is 15 in. Find the length of a radius of the circle. Round to the nearest hundredth. 53. Carpentry Find the length of molding needed to put around a circular table that is 4.2 ft in diameter. Round to the nearest hundredth. 54. Carpeting How much binding is needed to bind the edge of a circular rug that is 3 m in diameter? Round to the nearest hundredth. 55. Cycling A bicycle tire has a diameter of 24 in. How many feet does the bicycle travel when the wheel makes eight revolutions? Round to the nearest hundredth.

24 in.

56. Cycling A tricycle tire has a diameter of 12 in. How many feet does the tricycle travel when the wheel makes twelve revolutions? Round to the nearest hundredth. 57. Earth Science The distance from the surface of Earth to its center is 6356 km. What is the circumference of Earth? Round to the nearest hundredth.

Objective B

To solve problems involving the area of a geometric figure

Find the area of the figure. 59.

60. 5 ft

61. 6m

4.5 in.

12 ft 8m 4.5 in.


58. Sewing Bias binding is to be sewed around the edge of a rectangular tablecloth measuring 72 in. by 45 in. If the bias binding comes in packages containing 15 ft of binding, how many packages of bias binding are needed for the tablecloth?


62.

63.

64.

12 cm

26 ft 12 in.

8 cm 42 ft

20 in.

16 cm

Find the area of the figure. Give both the exact value and an approximation to the nearest hundredth. 65.

4 cm

68.

66.

67. 12 m

69.

5.5 mi

70.

18 in.

17 ft 6.6 km

Solve. 71. The length of a side of a square is 12.5 cm. Find the area of the square.


72. Each side of a square measures 3

1 2

in. Find the area of the square.

73. The length of a rectangle is 38 in., and the width is 15 in. Find the area of the rectangle. 74. Find the area of a rectangle that has a length of 6.5 m and a width of 3.8 m. 75. The length of the base of a parallelogram is 16 in., and the height is 12 in. Find the area of the parallelogram. 76. The height of a parallelogram is 3.4 m, and the length of the base is 5.2 m. Find the area of the parallelogram.

191

192


77. The length of the base of a triangle is 6 ft. The height is 4.5 ft. Find the area of the triangle. 78. The height of a triangle is 4.2 cm. The length of the base is 5 cm. Find the area of the triangle. 79. The length of one base of a trapezoid is 35 cm, and the length of the other base is 20 cm. If the height is 12 cm, what is the area of the trapezoid? 80. The height of a trapezoid is 5 in. The bases measure 16 in. and 18 in. Find the area of the trapezoid. 81. The radius of a circle is 5 in. Find the area of the circle. Give the exact value. 82. Find the area of a circle with a radius of 14 m. Round to the nearest hundredth. 83. Find the area of a circle that has a diameter of 3.4 ft. Round to the nearest hundredth. 84. The diameter of a circle is 6.5 m. Find the area of the circle. Give the exact value. 85. Telescopes The dome of the Hale telescope at Mount Palomar, California, has a diameter of 200 in. Find the area across the dome. Give the exact value. 86. Patios What is the area of a square patio that measures 8.5 m on each side?

88. Irrigation An irrigation system waters a circular field that has a 50-foot radius. Find the area watered by the irrigation system. Give the exact value. 89. Athletic Fields Artificial turf is being used to cover a playing field. If the field is rectangular with a length of 100 yd and a width of 75 yd, how much artificial turf must be purchased to cover the field? 90. Interior Decorating A fabric wall hanging is to fill a space that measures 5 m by 3.5 m. Allowing for 0.1 m of the fabric to be folded back along each edge, how much fabric must be purchased for the wall hanging? 91. The area of a rectangle is 300 in2. If the length of the rectangle is 30 in., what is the width?

30 in. W


87. Gardens Find the area of a rectangular flower garden that measures 14 ft by 9 ft.


92. The width of a rectangle is 12 ft. If the area is 312 ft2, what is the length of the rectangle? 93. The height of a triangle is 5 m. The area of the triangle is 50 m2. Find the length of the base of the triangle. 94. The area of a parallelogram is 42 m2. If the height of the parallelogram is 7 m, what is the length of the base? 95. Home Maintenance You plan to stain the wooden deck attached to your house. The deck measures 10 ft by 8 ft. If a quart of stain will cover 50 ft2, how many quarts of stain should you buy? 96. Flooring

You want to tile your kitchen floor. The floor measures 1 2

12 ft by 9 ft. How many tiles, each a square with side 1 ft, should you purchase for the job? 97. Interior Decorating You are wallpapering two walls of a child’s room, one measuring 9 ft by 8 ft and the other measuring 11 ft by 8 ft. The wallpaper costs $24.50 per roll, and each roll of the wallpaper will cover 40 ft2. What will it cost to wallpaper the two walls? 98. Parks An urban renewal project involves reseeding a park that is in the shape of a square 60 ft on each side. Each bag of grass seed costs $9.75 and will seed 1200 ft2. How much money should be budgeted for buying grass seed for the park? 99. A circle has a radius of 8 in. Find the increase in area when the radius is increased by 2 in. Round to the nearest hundredth.


100. A circle has a radius of 6 cm. Find the increase in area when the radius is doubled. Round to the nearest hundredth. 101. Carpeting You want to install wall-to-wall carpeting in your living room, which measures 15 ft by 24 ft. If the cost of the carpet you would like to purchase is $21.95 per square yard, what will be the cost of the carpeting for your living room? (Hint: 9 ft2 1 yd2) 102. Interior Decorating You want to paint the walls of your bedroom. Two walls measure 15 ft by 9 ft, and the other two walls measure 12 ft by 9 ft. The paint you wish to purchase costs $12.98 per gallon, and each gallon will cover 400 ft2 of wall. Find the total amount you will spend on paint. 103. Landscaping A walkway 2 m wide surrounds a rectangular plot of grass. The plot is 30 m long and 20 m wide. What is the area of the walkway?

193

194


104. Interior Decorating Pleated draperies for a window must be twice as wide as the width of the window. Draperies are being made for four windows, each 2 ft wide and 4 ft high. Because the drapes will fall slightly below the window sill, and because extra fabric will be needed for hemming the drapes, 1 ft must be added to the height of the window. How much material must be purchased to make the drapes? 105. Construction Find the cost of plastering the walls of a room 22 ft long, 25 ft 6 in. wide, and 8 ft high. Subtract 120 ft2 for windows and doors. The cost is $2.50 per square foot. APPLYING THE CONCEPTS 106. If both the length and the width of a rectangle are doubled, how many times larger is the area of the resulting rectangle? 107. A hexagram is a six-pointed star formed by extending each of the sides of a regular hexagon into an equilateral triangle. A hexagram is shown at the right. Use a pencil, a paper, a protractor, and a ruler to create a hexagram. C d

108. If the formula C d is solved for , the resulting equation is . Therefore, is the ratio of the circumference of a circle to the length of its diameter. Use several circular objects, such as coins, plates, tin cans, and wheels, to show that the ratio of the circumference of each object to its diameter is approximately 3.14.

110. Determine whether the statement is always true, sometimes true, or never true. a. If two triangles have the same perimeter, then they have the same area. b. If two rectangles have the same area, then they have the same perimeter. c. If two squares have the same area, then the sides of the squares have the same length. d. An equilateral triangle is also an isosceles triangle. e. All the radii (plural of radius) of a circle are equal. f. All the diameters of a circle are equal. 111.

Suppose a circle is cut into 16 equal pieces, which are then arranged as shown at the right. The figure formed resembles a parallelogram. What variable expression could describe the base of the parallelogram? What variable could describe its height? Explain how the formula for the area of a circle is derived from this approach.

112.

The apothem of a regular polygon is the distance from the center of the polygon to a side. Explain how to derive a formula for the area of a regular polygon using the apothem. (Hint: Use the formula for the area of a triangle.)

apothem


109. Derive a formula for the area of a circle in terms of the diameter of the circle.

195

Section 3.3 / Solids

3.3 Objective A

Solids To solve problems involving the volume of a solid

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SSM

Geometric solids are figures in space. Five common geometric solids are the rectangular solid, the sphere, the cylinder, the cone, and the pyramid. A rectangular solid is one in which all six sides, called faces, are rectangles. The variable L is used to represent the length of a rectangular solid, W its width, and H its height. A sphere is a solid in which all points are the same distance from point O, which is called the center of the sphere. The diameter, d, of a sphere is a line across the sphere going through point O. The radius, r, is a line from the center to a point on the sphere. AB is a diameter and OC is a radius of the sphere shown at the right.

H

W L C

A

B

O

d 2r or r The most common cylinder, called a right circular cylinder, is one in which the bases are circles and are perpendicular to the height of the cylinder. The variable r is used to represent the radius of a base of a cylinder, and h represents the height. In this text, only right circular cylinders are discussed.

h


r

A right circular cone is obtained when one base of a right circular cylinder is shrunk to a point, called a vertex, V. The variable r is used to represent the radius of the base of the cone, and h represents the height. The variable is used to represent the slant height, which is the distance from a point on the circumference of the base to the vertex. In this text, only right circular cones are discussed.

The base of a regular pyramid is a regular polygon, and the sides are isosceles triangles. The height, h, is the distance from the vertex, V, to the base and is perpendicular to the base. The variable is used to represent the slant height, which is the height of one of the isosceles triangles on the face of the pyramid. The regular square pyramid at the right has a square base. This is the only type of pyramid discussed in this text.

V

h

r

V

h Base s

s

1 d 2


A cube is a special type of rectangular solid. Each of the six faces of a cube is a square. The variable s is used to represent the length of one side of a cube.

s

s

s

Volume is a measure of the amount of space inside a figure in space. Volume can be used to describe the amount of heating gas used for cooking, the amount of concrete delivered for the foundation of a house, or the amount of water in storage for a city’s water supply. A cube that is 1 ft on each side has a volume of 1 cubic foot, which is written 1 ft3. A cube that measures 1 cm on each side has a volume of 1 cubic centimeter, which is written 1 cm3.

1 ft 1 ft

1 ft

1 cm 1 cm

The volume of a solid is the number of cubes that are necessary to exactly fill the solid. The volume of the rectangular solid at the right is 24 cm3 because it will hold exactly 24 cubes, each 1 cm on a side. Note that the volume can be found by multiplying the length times the width times the height.

1 cm

2 cm

3 cm 4 cm 4 . 3 . 2 = 24

The formulas for the volumes of the geometric solids described above are given below.

Volumes of Geometric Solids

The volume, V, of a rectangular solid with length L, width W, and height H is given by V LWH. The volume, V, of a cube with side s is given by V s 3. 4 3

The volume, V, of a sphere with radius r is given by V r 3. The volume, V, of a right circular cylinder is given by V r 2h, where r is the radius of the base and h is the height. 1

The volume, V, of a right circular cone is given by V r 2h, where r is the radius of the 3 circular base and h is the height. 1

The volume, V, of a regular square pyramid is given by V s 2h, where s is the length of a 3 side of the base and h is the height.


196


HOW TO

Integrating

Technology To approximate 36 on your calculator, enter 36 .

Find the volume of a sphere with a diameter of 6 in.

1 1 d (6) 3 2 2

• First find the radius of the sphere.

V

4 3 r 3

• Use the formula for the volume of a sphere.

V

4 (3)3 3

V

4 (27) 3

r

197

V 36

• The exact volume of the sphere is 36 p in3.

V 113.10

• An approximate measure can be found by using the p key on a calculator.

The approximate volume is 113.10 in3.

Example 1

You Try It 1

The length of a rectangular solid is 5 m, the width is 3.2 m, and the height is 4 m. Find the volume of the solid.

Find the volume of a cube that measures 2.5 m on a side.

Strategy

Your strategy

To find the volume, use the formula for the volume of a rectangular solid. L 5, W 3.2, H 4 Solution

Your solution

V LWH 5(3.2)(4) 64


The volume of the rectangular solid is 64 m3. Example 2

You Try It 2

The radius of the base of a cone is 8 cm. The height is 12 cm. Find the volume of the cone. Round to the nearest hundredth.

The diameter of the base of a cylinder is 8 ft. The height of the cylinder is 22 ft. Find the exact volume of the cylinder.

Strategy

Your strategy

To find the volume, use the formula for the volume of a cone. An approximation is asked for; use the key on a calculator. r 8, h 12 Solution

Your solution

1 V r2h 3 V

1 1 (8)2(12) (64)(12) 256 3 3

804.25 The volume is approximately 804.25 cm3. Solutions on pp. S10– S11


Objective B

To solve problems involving the surface area of a solid

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The surface area of a solid is the total area on the surface of the solid. When a rectangular solid is cut open and flattened out, each face is a rectangle. The surface area, SA, of the rectangular solid is the sum of the areas of the six rectangles:

H

W L

SA LW LH WH LW WH LH

LW LH

which simplifies to WH

SA 2LW 2LH 2WH

LW

WH

LH

The surface area of a cube is the sum of the areas of the six faces of the cube. The area of each face is s2. Therefore, the surface area, SA, of a cube is given by the formula SA 6s2.

s

s s

When a cylinder is cut open and flattened out, the top and bottom of the cylinder are circles. The side of the cylinder flattens out to a rectangle. The length of the rectangle is the circumference of the base, which is 2 r; the width is h, the height of the cylinder. Therefore, the area of the rectangle is 2 rh. The surface area, SA, of the cylinder is r

SA r 2 2 rh r 2 2πr

which simplifies to h

SA 2 r 2 2 rh r

h


198


The surface area of a regular square pyramid is the area of the base plus the area of the four isosceles triangles. A side of the square base is s; therefore, the area of the base is s2. The slant height, , is the height of each triangle, and s is the base of each triangle. The surface area, SA, of a pyramid is

SA s2 4

199

Base s

1 s 2

which simplifies to SA s2 2s

Formulas for the surface areas of geometric solids are given below.

Surface Areas of Geometric Solids

The surface area, SA, of a rectangular solid with length L, width W, and height H is given by SA 2LW 2LH 2WH. The surface area, SA, of a cube with side s is given by SA 6s 2. The surface area, SA, of a sphere with radius r is given by SA 4r 2. The surface area, SA, of a right circular cylinder is given by SA 2r 2 2rh, where r is the radius of the base and h is the height. The surface area, SA, of a right circular cone is given by SA r 2 r, where r is the radius of the circular base and is the slant height.


The surface area, SA, of a regular square pyramid is given by SA s 2 2s, where s is the length of a side of the base and is the slant height.

HOW TO r

Find the surface area of a sphere with a diameter of 18 cm.

1 1 d (18) 9 2 2

SA 4 r2 SA 4 (9)2

• First find the radius of the sphere. • Use the formula for the surface area of a sphere.

SA 4 (81) SA 324

• The exact surface area of the sphere is 324 p cm2.

SA 1017.88

• An approximate measure can be found by using the p key on a calculator.

The approximate surface area is 1017.88 cm2.

s

200


Example 3

You Try It 3

The diameter of the base of a cone is 5 m, and the slant height is 4 m. Find the surface area of the cone. Give the exact measure.

The diameter of the base of a cylinder is 6 ft, and the height is 8 ft. Find the surface area of the cylinder. Round to the nearest hundredth.

Strategy

Your strategy

To find the surface area of the cone: • Find the radius of the base of the cone. • Use the formula for the surface area of a cone. Leave the answer in terms of .

Solution

Your solution

1 1 r d (5) 2.5 2 2 SA SA SA SA SA

r 2 r (2.5)2 (2.5)(4) (6.25) (2.5)(4) 6.25 10 16.25

The surface area of the cone is 16.25 m2.

Example 4

You Try It 4

Find the area of a label used to cover a soup can that has a radius of 4 cm and a height of 12 cm. Round to the nearest hundredth.

Which has a larger surface area, a cube with a side measuring 10 cm or a sphere with a diameter measuring 8 cm?

Strategy

Your strategy

Solution

Your solution

Area of the label 2 rh Area of the label 2 (4)(12) 96 301.59 The area is approximately 301.59 cm2.

Solutions on p. S11


To find the area of the label, use the fact that the surface area of the sides of a cylinder is given by 2 rh. An approximation is asked for; use the key on a calculator. r 4, h 12



To solve problems involving the volume of a solid

Find the volume of the figure. For calculations involving , give both the exact value and an approximation to the nearest hundredth. 1.

2.

3.

6 in. 14 in.

14 ft

5 ft

10 in.

3 ft

12 ft

3 ft

4.

5.

6.

7.5 m 3 cm 7.5 m

8 cm

7.5 m 8 cm

Solve. 7. Storage Units A rectangular storage unit has a length of 6.8 m, a width of 2.5 m, and a height of 2 m. Find the volume of the storage unit. 8. Fish Hatchery A rectangular tank at a fish hatchery is 9 m long, 3 m wide, and 1.5 m deep. Find the volume of the water in the tank when the tank is full. 9. Find the volume of a cube whose side measures 3.5 in. 10. The length of a side of a cube is 7 cm. Find the volume of the cube.


11. The diameter of a sphere is 6 ft. Find the volume of the sphere. Give the exact measure. 12. Find the volume of a sphere that has a radius of 1.2 m. Round to the nearest tenth. 13. The diameter of the base of a cylinder is 24 cm. The height of the cylinder is 18 cm. Find the volume of the cylinder. Round to the nearest hundredth. 14. The height of a cylinder is 7.2 m. The radius of the base is 4 m. Find the volume of the cylinder. Give the exact measure. 15. The radius of the base of a cone is 5 in. The height of the cone is 9 in. Find the volume of the cone. Give the exact measure.

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202


16. The height of a cone is 15 cm. The diameter of the cone is 10 cm. Find the volume of the cone. Round to the nearest hundredth. 17. The length of a side of the base of a pyramid is 6 in., and the height is 10 in. Find the volume of the pyramid. 18. The height of a pyramid is 8 m, and the length of a side of the base is 9 m. What is the volume of the pyramid? 19. Appliances The volume of a freezer with a length of 7 ft and a height of 3 ft is 52.5 ft3. Find the width of the freezer. 20. Aquariums The length of an aquarium is 18 in., and the width is 12 in. If the volume of the aquarium is 1836 in3, what is the height of the aquarium? 21. The volume of a cylinder is 502.4 in3. The diameter of the base is 10 in. Find the height of the cylinder. Round to the nearest hundredth. 22. The diameter of the base of a cylinder is 14 cm. If the volume of the cylinder is 2310 cm3, find the height of the cylinder. Round to the nearest hundredth. 23. A rectangular solid has a square base and a height of 5 in. If the volume of the solid is 125 in3, find the length and the width. 24. The volume of a rectangular solid is 864 m3. The rectangular solid has a square base and a height of 6 m. Find the dimensions of the solid. 25. Petroleum An oil storage tank, which is in the shape of a cylinder, is 4 m high and has a diameter of 6 m. The oil tank is two-thirds full. Find the number of cubic meters of oil in the tank. Round to the nearest hundredth.

Objective B

To solve problems involving the surface area of a solid

Find the surface area of the figure. 27.

28.

29. 5m

3m 14 ft 4m

5m 14 ft

14 ft

4m

4m


26. Agriculture A silo, which is in the shape of a cylinder, is 16 ft in diameter and has a height of 30 ft. The silo is three-fourths full. Find the volume of the portion of the silo that is not being used for storage. Round to the nearest hundredth.


Find the surface area of the figure. Give both the exact value and an approximation to the nearest hundredth. 30.

31.

32. 2 in.

2 cm

9 ft

6 in.

3 ft

Solve. 33. The height of a rectangular solid is 5 ft. The length is 8 ft, and the width is 4 ft. Find the surface area of the solid. 34. The width of a rectangular solid is 32 cm. The length is 60 cm, and the height is 14 cm. What is the surface area of the solid? 35. The side of a cube measures 3.4 m. Find the surface area of the cube. 36. Find the surface area of a cube that has a side measuring 1.5 in. 37. Find the surface area of a sphere with a diameter of 15 cm. Give the exact value. 38. The radius of a sphere is 2 in. Find the surface area of the sphere. Round to the nearest hundredth. 39. The radius of the base of a cylinder is 4 in. The height of the cylinder is 12 in. Find the surface area of the cylinder. Round to the nearest hundredth.


40. The diameter of the base of a cylinder is 1.8 m. The height of the cylinder is 0.7 m. Find the surface area of the cylinder. Give the exact value. 41. The slant height of a cone is 2.5 ft. The radius of the base is 1.5 ft. Find the surface area of the cone. Give the exact value. 42. The diameter of the base of a cone is 21 in. The slant height is 16 in. What is the surface area of the cone? Round to the nearest hundredth. 43. The length of a side of the base of a pyramid is 9 in., and the slant height is 12 in. Find the surface area of the pyramid. 44. The slant height of a pyramid is 18 m, and the length of a side of the base is 16 m. What is the surface area of the pyramid?

203

204


45. The surface area of a rectangular solid is 108 cm2. The height of the solid is 4 cm, and the length is 6 cm. Find the width of the rectangular solid. 46. The length of a rectangular solid is 12 ft. The width is 3 ft. If the surface area is 162 ft2, find the height of the rectangular solid. 47. Paint A can of paint will cover 300 ft2. How many cans of paint should be purchased in order to paint a cylinder that has a height of 30 ft and a radius of 12 ft? 48. Ballooning A hot air balloon is in the shape of a sphere. Approximately how much fabric was used to construct the balloon if its diameter is 32 ft? Round to the nearest whole number. 49. Aquariums How much glass is needed to make a fish tank that is 12 in. long, 8 in. wide, and 9 in. high? The fish tank is open at the top. 50. Packaging Find the area of a label used to cover a can of juice that has a diameter of 16.5 cm and a height of 17 cm. Round to the nearest hundredth. 51. The length of a side of the base of a pyramid is 5 cm, and the slant height is 8 cm. How much larger is the surface area of this pyramid than the surface area of a cone with a diameter of 5 cm and a slant height of 8 cm? Round to the nearest hundredth.

APPLYING THE CONCEPTS 52. Half of a sphere is called a hemisphere. Derive formulas for the volume and surface area of a hemisphere. 53. Determine whether the statement is always true, sometimes true, or never true. a. The slant height of a regular pyramid is longer than the height.

54. a. What is the effect on the surface area of a rectangular solid when the width and height are doubled? b. What is the effect on the volume of a rectangular solid when both the length and the width are doubled? c. What is the effect on the volume of a cube when the length of each side of the cube is doubled? d. What is the effect on the surface area of a cylinder when the radius and height are doubled? 55. a. b. c. d.

Explain how you could cut through a cube so that the face of the resulting solid is a square an equilateral triangle a trapezoid a hexagon


b. The slant height of a cone is shorter than the height. c. The four triangular faces of a regular pyramid are equilateral triangles.


205

Focus on Problem Solving More on the Trial-and-Error Approach to Problem Solving

Some problems in mathematics are solved by using trial and error. The trialand-error method of arriving at a solution to a problem involves repeated tests or experiments until a satisfactory conclusion is reached. Many of the Applying the Concepts exercises in this text require a trial-and-error method of solution. For example, an exercise in Section 3 of this chapter reads: Explain how you could cut through a cube so that the face of the resulting solid is (a) a square, (b) an equilateral triangle, (c) a trapezoid, (d) a hexagon. There is no formula to apply to this problem; there is no computation to perform. This problem requires picturing a cube and the results after cutting through it at different places on its surface and at different angles. For part (a), cutting perpendicular to the top and bottom of the cube and parallel to two of its sides will result in a square. The other shapes may prove more difficult.


When solving problems of this type, keep an open mind. Sometimes when using the trial-and-error method, we are hampered by narrowness of vision; we cannot expand our thinking to include other possibilities. Then when we see someone else’s solution, it appears so obvious to us! For example, for the Applying the Concepts question above, it is necessary to conceive of cutting through the cube at places other than the top surface; we need to be open to the idea of beginning the cut at one of the corner points of the cube. A topic of the Projects and Group Activities in this chapter is symmetry. Here again, trial and error is used to determine the lines of symmetry inherent in an object. For example, in determining lines of symmetry for a square, begin by drawing a square. The horizontal line of symmetry and the vertical line of symmetry may be immediately obvious to you. But there are two others. Do you see that a line drawn through opposite corners of the square is also a line of symmetry? Many of the questions in this text that require an answer of “always true, sometimes true, or never true” are best solved by the trial-and-error method. For example, consider the statement presented in Section 2 of this chapter. If two rectangles have the same area, then they have the same perimeter.


Try some numbers. Each of two rectangles, one measuring 6 units by 2 units and another measuring 4 units by 3 units, has an area of 12 square units, but the perimeter of the first is 16 units and the perimeter of the second is 14 units. So the answer “always true” has been eliminated. We still need to determine whether there is a case when the statement is true. After experimenting with a lot of numbers, you may come to realize that we are trying to determine if it is possible for two different pairs of factors of a number to have the same sum. Is it? Don’t be afraid to make many experiments, and remember that errors, or tests that “don’t work,” are a part of the trial-and-error process.

Projects and Group Activities Investigating Perimeter

The perimeter of the square at the right is 4 units. If two squares are joined along one of the sides, the perimeter is 6 units. Note that it does not matter which sides are joined; the perimeter is still 6 units. If three squares are joined, the perimeter of the resulting figure is 8 units for each possible placement of the squares. Four squares can be joined in five different ways as shown. There are two possible perimeters, 10 units for A, B, C, and D, and 8 units for E.

A

C

1. If five squares are joined, what is the maximum perimeter possible? 2. If five squares are joined, what is the minimum perimeter possible? 3. If six squares are joined, what is the maximum perimeter possible? 4. If six squares are joined, what is the minimum perimeter possible?

B

D

E


206

Chapter 3 Summary

Symmetry

207

Look at the letter A printed at the left. If the letter were folded along line , the two sides of the letter would match exactly. This letter has symmetry with respect to line . Line is called the axis of symmetry. Now consider the letter H printed below at the left. Both lines 1 and 2 are axes of symmetry for this letter; the letter could be folded along either line and the two sides would match exactly. 1. Does the letter A have more than one axis of symmetry? 2. Find axes of symmetry for other capital letters of the alphabet. 3. Which lowercase letters have one axis of symmetry? 4. Do any of the lowercase letters have more than one axis of symmetry?

1

2

5. Find the number of axes of symmetry for each of the plane geometric figures presented in this chapter. 6. There are other types of symmetry. Look up the meaning of point symmetry and rotational symmetry. Which plane geometric figures provide examples of these types of symmetry? 7. Find examples of symmetry in nature, art, and architecture.


Examples

A line extends indefinitely in two directions. A line segment is part of a line and has two endpoints. The length of a line segment is the distance between the endpoints of the line segment. [3.1A, p. 161]

Line


A

Parallel lines never meet; the distance between them is always the same. The symbol!means “is parallel to.” Intersecting lines cross at a point in the plane. Perpendicular lines are intersecting lines that form right angles. The symbol means “is perpendicular to.” [3.1A, pp. 162–163]

Line B segment

Parallel lines

Perpendicular lines

208


A ray starts at a point and extends indefinitely in one direction. An angle is formed when two rays start from the same point. The common point is called the vertex of the angle. An angle is measured in degrees. A 90° angle is a right angle. A 180° angle is a straight angle. [3.1A, pp. 161 –164]

90°

180° A

Complementary angles are two angles whose measures have the sum 90°. Supplementary angles are two angles whose measures have the sum 180°. [3.1A, pp. 163 – 164]

Right angle

O B Straight angle

130° 50° B

A

An acute angle is an angle whose measure is between 0° and 90°. An obtuse angle is an angle whose measure is between 90° and 180°. [3.1A, p. 164]

A above is an obtuse angle. B above is an acute angle.

Two angles that are on opposite sides of the intersection of two lines are vertical angles; vertical angles have the same measure. Two angles that share a common side are adjacent angles; adjacent angles of intersecting lines are supplementary angles. [3.1B, p. 166]

x

p y

w

q

z

Angles w and y are vertical angles. Angles x and y are adjacent angles.

t b

a d w z

1

c x

2

y

Parallel lines 1 and 2 are cut by transversal t. All four acute angles have the same measure. All four obtuse angles have the same measure.

A quadrilateral is a four-sided polygon. A parallelogram, a rectangle, and a square are quadrilaterals. [3.2A, pp. 177 –178] Rectangle

Square


A line that intersects two other lines at two different points is a transversal. If the lines cut by a transversal are parallel lines, equal angles are formed: alternate interior angles, alternate exterior angles, and corresponding angles. [3.1B, p. 167]

Chapter 3 Summary

A polygon is a closed figure determined by three or more line segments. The line segments that form the polygon are its sides. A regular polygon is one in which each side has the same length and each angle has the same measure. Polygons are classified by the number of sides. [3.2A, p. 177]

Number of Sides

Name of the Polygon

3 4 5 6 7 8 9 10

Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon

A triangle is a closed, three-sided plane figure. [3.1C, p. 169] An isosceles triangle has two sides of equal length. The three sides of an equilateral triangle are of equal length. A scalene triangle has no two sides of equal length. An acute triangle has three actue angles. An obtuse triangle has one obtuse angle. A right triangle contains a right angle. [3.2A, pp. 177 – 178] A circle is a plane figure in which all points are the same distance from the center of the circle. A diameter of a circle is a line segment across the circle through the center. A radius of a circle is a line segment from the center of the circle to a point on the circle. [3.2A, p. 180]

209

Right Triangle

C

A

O

B

AB is a diameter of the circle. OC is a radius.


Geometric solids are figures in space. Six common space figures are the rectangular solid, cube, sphere, cylinder, cone, and pyramid. A rectangular solid is a solid in which all six faces are rectangles. A cube is a rectangular solid in which all six faces are squares. A sphere is a solid in which all points on the sphere are the same distance from the center of the sphere. The most common cylinder is one in which the bases are circles and are perpendicular to the height. In this text the only types of cones and pyramids discussed are the right circular cone and the regular square pyramid. [3.3A, pp. 195 – 196]

Height Length

Wi

dth

Rectangular Solid

Essential Rules and Procedures

Examples

Triangles [3.1C, p. 169]

Two angles of a triangle measure 32° and 48°. Find the measure of the third angle.

Sum of the measures of the interior angles 180°

A B C 180° A 32° 48° 180° A 80° 180° A 80° 80° 180° 80° A 100° The measure of the third angle is 100°.

210


Formulas for Perimeter (the distance around a figure)

[3.2A, pp. 178–180] Triangle: P a b c Rectangle: P 2L 2W Square: P 4s Circumference of a circle: C d or C 2 r

Formulas for Area (the amount of surface in a region)

[3.2B, pp. 182 –185] Rectangle: A LW Square: A s2 Parallelogram: A bh 1 2

Triangle: A bh 1 2

Trapezoid: A hb1 b2

The length of a rectangle is 8 m. The width is 5.5 m. Find the perimeter of the rectangle. P 2L 2W P 28 25.5 P 16 11 P 27 The perimeter is 27 m.

Find the area of a circle with a radius of 4 cm. Round to the nearest hundredth. A r2 A 42 A 16 A 50.27 The area is approximately 50.27 cm2.

Circle: A r 2

Formulas for Volume (the amount of space inside a figure in space) [3.3A, p. 196] Rectangular solid: V LWH Cube: V s3 4 3

Sphere: V r 3 Cylinder: V r 2h

Find the volume of a cube that measures 3 in. on a side. V s3 V 33 V 27 The volume is 27 in3.

1 3

Right circular cone: V r 2h 1 3

Formulas for Surface Area (the total area on the surface of a

solid) [3.3B, p. 199] Rectangular solid: 2LW 2LH 2WH Cube: SA 6s2 Sphere: SA 4 r 2 Cylinder: SA 2 r 2 2 rh Right circular cone: SA r 2 rl Regular square pyramid: SA s2 2sl

Find the surface area of a sphere that has a radius of 3 in. Round to the nearest hundredth. SA 4 r 2 SA 432 SA 49 SA 36 SA 113.10 The surface area is approximately 113.10 in3.


Regular square pyramid: V s2h


211


Given that a 74° and b 52°, find the measures of angles x and y.

02. Find the measure of x. m

a b

112° x

x y

n

03. Given that BC 11 cm and AB is three times the length of BC, find the length of AC. A

B

04. Find x.

C 4x

05. Find the volume of the figure.

8 cm 6 cm

3x

x + 28°

06. Given that 1 2, find the measures of angles a and b.

138°

1

a

2

b

6 cm

t


07. Find the surface area of the figure.

08. Find the supplement of a 32° angle.

10 ft

4 ft 5 ft

09. Determine the area of a rectangle with a length of 12 cm and a width of 6.5 cm.

10. Determine the area of a triangle whose base is 9 m and whose height is 14 m.

212


11. Find the volume of a rectangular solid with a length of 6.5 ft, a width of 2 ft, and a height of 3 ft.

12. Two angles of a triangle measure 37° and 48°. Find the measure of the third angle.

13. The height of a triangle is 7 cm. The area of the triangle is 28 cm 2. Find the length of the base of the triangle.

14. Find the volume of a sphere that has a diameter of 12 mm. Find the exact value.

15. Determine the exact volume of a right circular cone whose radius is 7 cm and whose height is 16 cm.

16. Framing The perimeter of a square picture frame is 86 cm. Find the length of each side of the frame.

18. Parks The length of a rectangular park is 56 yd. The width is 48 yd. How many yards of fencing are needed to surround the park?

19. Patios What is the area of a square patio that measures 9.5 m on each side?

20. Landscaping A walkway 2 m wide surrounds a rectangular plot of grass. The plot is 40 m long and 25 m wide. What is the area of the walkway?


17. Paint A can of paint will cover 200 ft2. How many cans of paint should be purchased in order to paint a cylinder that has a height of 15 ft and a radius of 6 ft?

Chapter 3 Test

213

Chapter 3 Test 01. The diameter of a sphere is 1.5 m. Find the radius of the sphere.

02. Find the circumference of a circle with a radius of 5 cm. Round to the nearest hundredth.

03. Find the perimeter of the rectangle in the figure below.

04. Given AB 15, CD 6, and AD 24, find the length of BC. A

B C

D

5 ft 8 ft

05. Find the volume of a sphere with a diameter of 8 ft. Round to the nearest hundredth.

06. Find the area of the circle shown below. Round to the nearest hundredth.

9 cm


08. Find the supplement of a 105° angle.

t

1

a b

2


80°


10. Find the area of the rectangle shown below.

t

5m b

1

45°

a 2

11 m

214


11. Find the volume of a cylinder with a height of 6 m and a radius of 3 m. Round to the nearest hundredth.

12. Find the perimeter of a rectangle that has a length of 2 m and a width of 1.4 m.

13. Find the complement of a 32° angle.

14. Find the surface area of the figure. Round to the nearest hundredth.

8 ft

5 ft

15. Pizza How much more pizza is contained in a pizza with radius 10 in. than in one with radius 8 in.? Round to the nearest hundredth.

16. Triangles A right triangle has a 32° angle. Find the measures of the other two angles.

17. Cycling A bicycle tire has a diameter of 28 in. How many feet does the bicycle travel if the wheel makes 10 revolutions? Round to the nearest tenth. 28 in.

19. Agriculture A silo, which is in the shape of a cylinder, is 9 ft in diameter and has a height of 18 ft. Find the volume of the silo. Round to the nearest hundredth.

20. Triangles Find the area of a right triangle with a base of 8 m and a height of 2.75 m.


18. Carpeting New carpet is installed in a room measuring 18 ft by 14 ft. Find the area of the room in square yards. (9 ft2 1 yd2)


215

Cumulative Review Exercises 01. Let x {3, 0, 1}. For what values of x is the inequality x 1 a true statement?


03. Write

7 20

as a percent.

02. Write 8.9% as a decimal.

04. Divide:

4 9

2 3

05. Multiply: 5.7(4.3)

06. Simplify: 125

07. Evaluate 5 3[10 (5 6)2].

08. Evaluate a(b c)3 when a 1, b 2, and c 4.

09. Simplify: 5m 3n 8m

10. Simplify: 7(3y)

11. Simplify: 4(3x 2) (5x 1)

12. Use the roster method to write the set of negative integers greater than or equal to 2.

13. Find C D, given C {0, 10, 20, 30} and D {10, 0, 10}.

14. Graph: x 1 −5 −4 −3 −2 −1

0

1

2

3

4

5

15. Solve: 4x 2 6x 8

16. Solve: 3(2x 5) 18

17. Solve: 4y 3 6y 5

18. Solve: 8 4(3x 5) 6(x 8)

216


19. Solve: 2x 3 5 or x 4 1

20. Solve: 3 2x 7 5

21. Solve: 3x 1 2

22. Solve: x 8 2

23. Find the measure of x.

p

x 49°

q

24. Translate “the difference between four times a number and ten is two” into an equation and solve.

25. Triangles Two angles of a triangle measure 37° and 21°. Find the measure of the third angle of the triangle.

26. Investments Michael deposits $5000 in an account that earns an annual simple interest rate of 4.5% and $2500 in an account that earns an annual simple interest rate of 3.5%. How much interest will Michael earn from the two accounts in one year?

28.

Annual Earnings According to the Census Bureau, the median annual earnings of a man with a bachelor’s degree is $49,982, and the median earnings of a woman with a bachelor’s degree is $35,408. What percent of the men’s median annual earnings is the women’s median annual earnings? Round to the nearest tenth of a percent. (Median is a type of average.)

29. Find the exact area of a circle that has a diameter of 9 cm.

30. The volume of a box is 144 ft3. The length of the box is 12 ft, and the width is 4 ft. Find the height of the box.


27. Triangles Two sides of an isosceles triangle measure 7.5 m. The perimeter of the triangle is 19.5 m. Find the measure of the third side of the triangle.

chapter

4

Linear Functions and Inequalities in Two Variables

OBJECTIVES

Section 4.1

A B C

To graph points in a rectangular coordinate system To determine ordered-pair solutions of an equation in two variables To graph a scatter diagram

Section 4.2

A

To evaluate a function

Section 4.3


A B Do you have a cellular phone? Do you pay a monthly fee plus a charge for each minute you use? In this chapter you will learn how to write a linear function that models the monthly cost of a cell phone in terms of the number of minutes you use it. Exercise 78 on page 271 asks you to write linear functions for certain cell phone options. Being able to use a linear function to model a relationship between two variables is an important skill in many fields, such as business, economics, and nutrition.

C D

To graph a linear function To graph an equation of the form Ax By C To find the x- and y-intercepts of a straight line To solve application problems

Section 4.4

A B

To find the slope of a line given two points To graph a line given a point and the slope

Section 4.5

A B C

To find the equation of a line given a point and the slope To find the equation of a line given two points To solve application problems

Section 4.6

A

To find parallel and perpendicular lines

Section 4.7 Need help? For online student resources, such as section quizzes, visit this textbook’s Website at math.college.hmco.com/students.

A

To graph the solution set of an inequality in two variables

PREP TEST Do these exercises to prepare for Chapter 4.

1.

4x 3

2.

62 82

3.

3 5 26

4.

Evaluate 2x 5 for x 3.

5.

Evaluate

2r for r 5 . r1

6.

Evaluate 2p3 3p 4 for p 1.

7.

Evaluate

x1 x2 for x1 7 and x2 5. 2

8.

Given 3x 4y 12, find the value of x when y 0.

GO FIGURE If 5 4 and 5 6 and y x 1, which of the following has the largest value? x x y y


For Exercises 1 to 3, simplify.

Section 4.1 / The Rectangular Coordinate System

4.1 Objective A

Point of Interest A rectangular coordinate system is also called a Cartesian coordinate system, in honor of Descartes.

219

The Rectangular Coordinate System To graph points in a rectangular coordinate system

VIDEO & DVD

CD TUTOR

SSM

WEB

Before the 15th century, geometry and algebra were considered separate branches of mathematics. That all changed when René Descartes, a French mathematician who lived from 1596 to 1650, founded analytic geometry. In this geometry, a coordinate system is used to study relationships between variables.

A rectangular coordinate system is formed by two number lines, one horizontal and one vertical, that intersect at the zero point of each line. The point of intersection is called the origin. The two lines are called coordinate axes, or simply axes.

y Quadrant II Quadrant I horizontal axis

vertical axis x origin

The axes determine a plane, which can be thought of as a large, flat sheet of paper. The two axes divide the plane into four regions called quadrants. The quadrants are numbered counterclockwise from I to IV.

Point of Interest Gottfried Leibnitz introduced the words abscissa and ordinate. Abscissa is from Latin, meaning “to cut off.” Originally, Leibnitz used the phrase abscissa linea, “cut off a line” (axis). The root of ordinate is also a Latin word used to suggest a sense of order.

Quadrant IV

Each point in the plane can be identified by a pair of numbers called an ordered pair. The first number of the pair measures a horizontal distance and is called the abscissa. The second number of the pair measures a vertical distance and is called the ordinate. The coordinates of a point are the numbers in the ordered pair associated with the point. The abscissa is also called the first coordinate of the ordered pair, and the ordinate is also called the second coordinate of the ordered pair.

Horizontal distance Copyright © Houghton Mifflin Company. All rights reserved.

Quadrant III

Ordered pair Abscissa

Vertical distance (2, 3) Ordinate

Graphing, or plotting, an ordered pair in the plane means placing a dot at the location given by the ordered pair. The graph of an ordered pair is the dot drawn at the coordinates of the point in the plane. The points whose coordinates are (3, 4) and 2.5, 3) are graphed in the figure at the right.

y 4

(3, 4) 4 up

2

2.5 left 3 right –4

–2

3 down

0 –2

(−2.5, −3) – 4

2

4

x

220


The points whose coordinates are 3, 1 and 1, 3 are graphed at the right. Note that the graphed points are in different locations. The order of the coordinates of an ordered pair is important.

TA K E N O T E This is very important. An ordered pair is a pair of coordinates, and the order in which the coordinates appear is crucial.

y 4

(−1, 3)

2 −4

−2

0

4

2

x

(3, −1)

−2 −4

y

Each point in the plane is associated with an ordered pair, and each ordered pair is associated with a point in the plane. Although only the labels for integers are given on a coordinate grid, the graph of any ordered pair can be approximated. For example, the points whose coordinates are 2.3, 4.1 and , 1 are shown on the graph at the right.

Graph the ordered pairs 2, 3, 3, 2, 0, 2, and 3, 0. y

Solution

−4

Example 2

−2

You Try It 1

2 −2

−4

2 −4

−2

0

−2

−2

−4

−4

Give the coordinates of the points labeled A and B. Give the abscissa of point C and the ordinate of point D.

You Try It 2

C

−4

Solution

4

0

2

4

x

y B

B

2 −2

x

Give the coordinates of the points labeled A and B. Give the abscissa of point D and the ordinate of point C.

y

A

4

y

2

x

2

Graph the ordered pairs 4, 1, 3, 3, 0, 4, and 3, 0.

4

4

0

−4

Your solution

2

(π, 1)

−2

4

0

4

2

D 2

4

4

x

−4

C −2

0

−2

−2

−4

−4

The coordinates of A are 4, 2. The coordinates of B are 4, 4. The abscissa of C is 1. The ordinate of D is 1.

2

4

x

A D

Your solution

Solutions on p. S11


Example 1

(−2.3, 4.1)


Objective B

To determine ordered-pair solutions of an equation in two variables

VIDEO & DVD

CD TUTOR

WEB

221

SSM

When drawing a rectangular coordinate system, we often label the horizontal axis x and the vertical axis y. In this case, the coordinate system is called the xy-coordinate system. The coordinates of the points are given by ordered pairs (x, y), where the abscissa is called the x-coordinate and the ordinate is called the y-coordinate. A coordinate system is used to study the relationship between two variables. Frequently this relationship is given by an equation. Examples of equations in two variables include y 2x 3

3x 2y 6

x2 y 0

A solution of an equation in two variables is an ordered pair (x, y) whose coordinates make the equation a true statement. TA K E N O T E An ordered pair is of the form x, y. For the ordered pair 3, 7, 3 is the x value and 7 is the y value. Substitute 3 for x and 7 for y.

HOW TO

Is the ordered pair (3, 7) a solution of the equation y 2x 1?

y 2x 1 7 2(3) 1 761 77

• Replace x by 23 and y by 7. • Simplify. • Compare the results. If the resulting

Yes, the ordered pair (3, 7) is a solution of the equation.

equation is a true statement, the ordered pair is a solution of the equation. If it is not a true statement, the ordered pair is not a solution of the equation.

Besides (3, 7), there are many other ordered-pair solutions of y 2x 1. For example, (0, 1),

, 4, and (4, 7) are also solutions.

3 2


In general, an equation in two variables has an infinite number of solutions. By choosing any value of x and substituting that value into the equation, we can calculate a corresponding value of y.

HOW TO

2 3

Find the ordered-pair solution of y x 3 that corresponds to

x 6. 2 x3 3 2 (6) 3 3 43 1

y

• Replace x by 6. • Solve for y.

The ordered-pair solution is (6, 1). The solution of an equation in two variables can be graphed in an xy-coordinate system.

222


HOW TO Graph the ordered-pair solutions of y 2x 1 when x 2, 1, 0, 1, and 2. Use the values of x to determine ordered-pair solutions of the equation. It is convenient to record these in a table. x 2 1 0 1 2

y 2x 1 y y y y y

2(2) 1 2(1) 1 2(0) 1 2(1) 1 2(2) 1

y

(x, y)

5 3 1 1 3

(2, 5) (1, 3) (0, 1) (1, 1) (2, 3)

y 4 2 −4

−2

0

2

4

x

−2 −4

Example 3

You Try It 3

Is (3, 2) a solution of 3x 4y 15?

Is (2, 4) a solution of x 3y 14?

Solution

Your solution

3x 4y 15 3(3) 4(2) 15 • Replace x by 3 and y 9 8 15 by 22. 17 15 No, (3, 2) is not a solution of 3x 4y 15.

You Try It 4

Find the ordered-pair solution of

Find the ordered-pair solution of y

y

x x2

corresponding to x 4.

Solution

corresponding to x 2.

3x x1

Your solution

Replace x by 4 and solve for y. y

x 4 4 2 x2 42 2

The ordered-pair solution is (4, 2).

Solutions on p. S11


Example 4


Example 5

You Try It 5

Graph the ordered-pair solutions of

Graph the ordered-pair solutions of

y

2 x 3

2 when x 3, 0, 3, 6.

Solution

y

1 x 2

223

2 when x 4, 2, 0, 2.

Your solution

2 3

Replace x in y x 2 by 3, 0, 3, and 6. For each value of x, determine the value of y.

x

y

3

y

0 3 6

2 x2 3

2 (3) 2 3 2 y (0) 2 3 2 y (3) 2 3 2 y (6) 2 3

y

(x, y)

4

(3, 4)

2

(0, 2)

0

(3, 0)

2

(6, 2)

y

y

4

4 (6, 2)

2

(3, 0) −2

0 −2

2 (0, −2)

4

2

x

–4

–2

0

2

4

x

–2

(−3, −4) −4

–4


Solution on p. S12

Objective C

To graph a scatter diagram

VIDEO & DVD

CD TUTOR

WEB

SSM

Discovering a relationship between two variables is an important task in the study of mathematics. These relationships occur in many forms and in a wide variety of applications. Here are some examples. • A botanist wants to know the relationship between the number of bushels of wheat yielded per acre and the amount of watering per acre. • An environmental scientist wants to know the relationship between the incidence of skin cancer and the amount of ozone in the atmosphere. • A business analyst wants to know the relationship between the price of a product and the number of products that are sold at that price.

224


A researcher may investigate the relationship between two variables by means of regression analysis, which is a branch of statistics. The study of the relationship between the two variables may begin with a scatter diagram, which is a graph of the ordered pairs of the known data.

Integrating

Technology See the appendix Keystroke Guide: Scatter Diagrams for instructions on using a graphing calculator to create a scatter diagram.

The following table shows randomly selected data for a recent Boston Marathon. Ages of participants 40 years old and older and their times (in minutes) are given. Age (x)

55

46

53

40

40

44

54

44

41

50

Time (y) 254

204

243

194

281

197

238

300

232

216

y

The jagged portion of the horizontal axis in the figure at the right indicates that the numbers between 0 and 40 are missing.

Time (in minutes)

The scatter diagram for these data is shown at the right. Each ordered pair represents the age and time of a participant. For instance, the ordered pair (53, 243) indicates that a 53-year-old participant ran the marathon in 243 min.

TA K E N O T E

300 200 100 0

40

45

50

55

x

Age

Example 6

You Try It 6

The grams of sugar and the grams of fiber in a 1-ounce serving of six breakfast cereals are shown in the table below. Draw a scatter diagram of these data.

According to the National Interagency Fire Center, the number of deaths in U.S. wildland fires is as shown in the table below. Draw a scatter diagram of these data.

Sugar (x)

Fiber (y)

Year

Number of Deaths

4

3

1998

14

0

1999

28

3

2000

17

2

2001

18

1

2002

23

5

2003

29

Wheaties Rice Krispies

3

Total

5

Life

6

Kix

3

Grape-Nuts

7

Your strategy

To draw a scatter diagram: • Draw a coordinate grid with the horizontal axis representing the grams of sugar and the vertical axis the grams of fiber. • Graph the ordered pairs (4, 3), (3, 0), (5, 3), (6, 2), (3, 1), and (7, 5). Solution

Your solution

y

6 4 2 0

2

4

6

Grams of sugar

8

x

Number of deaths

Grams of fiber

8

y 30 25 20 15 10 5 0

'98 '99 '00 '01 '02 '03

x

Year

Solution on p. S12


Strategy

225



To graph points in a rectangular coordinate system

Graph 2, 1, 3, 5 , 2, 4, and 0, 3.

2. Graph 5, 1, 3, 3 , 1, 0, and 1, 1.

4.

–2

4

4

2

2

0

2

4

x

–4

–2

7.

5.

Graph 1, 4, 2, 3 , 0, 2, and 4, 0.

4

2

2

2

4

x

–4

–2

0

2

4

x

–2

–4

–4

–4

Find the coordinates of each of the points.

0

2

4

A A

–2

B

–2

0

2

4

x

–2

–2 –4

2

D

4

–2

0 –2 –4

x

–4

12. a. Name the abscissas of points A and C. b. Name the ordinates of points B and D. C

4

A

2

C –4

4

–2

2 B

x

2

y

4

A

2 0

0

y

C –2

–2

D

11. a. Name the abscissas of points A and C. b. Name the ordinates of points B and D.

4

B C

–4

D

10. Find the coordinates of each of the points.

y

4 2

C

x

x

y

4 2

4

9. Find the coordinates of each of the points.

y A

2

0

–2

–4

–4

–2

–2

8.

x

y

4

2

4

6. Graph 5, 2, 4, 1, 0, 0, and 0, 3.

4

0

2

0

y

–4

B

–2

–4

B


–4

–4

2

D

4

–4

4

–2

2

–2

y

–4

2

x

–2

Find the coordinates of each of the points.

C

4

0

y

–2

y

–2

Graph 4, 5, 3, 1 , 3, 4, and 5, 0.

–4

Graph 0, 0, 0, 5, 3, 0, and 0, 2.

y

y

–4

3.

2

D

4

x

B

A –4

D

–2

0 –2 –4

2

4

x

226 13.


Suppose you are helping a student who is having trouble graphing ordered pairs. The work of the student is at the right. What can you say to this student to correct the error that is being made?

y 4 (4, −3) (3, 2)

2

(0, 4)

14. a. What are the signs of the coordinates of a point in the third quadrant?

–4

–2

0

b. What are the signs of the coordinates of a point in the fourth quadrant?

–2

c. On an xy-coordinate system, what is the name of the axis for which all the x-coordinates are zero?

–4

2

4

x

(−3, 0)

d. On an xy-coordinate system, what is the name of the axis for which all the y-coordinates are zero?

Objective B

To determine ordered-pair solutions of an equation in two variables

15. Is 3, 4 a solution of y x 7?

16.

Is 2, 3 a solution of y x 5?

17. Is 1, 2 a solution of y x 1?

18.

Is 1, 3 a solution of y 2x 1?

19. Is 4, 1 a solution of 2x 5y 4?

20.

Is 5, 3 a solution of 3x 2y 9?

21. Is 0, 4 a solution of 3x 4y 4?

22.

Is 2, 0 a solution of x 2y 1?

1 2

23.

Find the ordered-pair solution of y 3x 2 corresponding to x 3.

24.


25.

Find the ordered-pair solution of y x 1 corresponding to x 6.

26.


27.


28.


29.


30.

Find the ordered-pair solution of y

2 3

2 5

2 5

1 x 6

2 corresponding to x 12.


3 4


Graph the ordered-pair solutions for the given values of x. 31.

y 2x; x 2, 1, 0, 2

32.

y 2x; x 2, 1, 0, 2

y

–4

33.

–2

y

4

4

2

2

0

2

4

x

–4

–2

0

–2

–2

–4

–4

y x 2; x 4, 2, 0, 3

34.

y

35.

y

–2

4

2

2 2

4

x

–4

–2

0

–2

–2

–4

–4

2 x 1; x 3, 0, 3 3

36.


37.

–2

4

2

2 2

4

x

–4

–2

0

–2

–2

–4

–4

y x2; x 2, 1, 0, 1, 2

38.

–4

12

4

8

–4 –8

2

4

x

y

8

0

x

y x 1; x 5, 3, 0, 3, 5

y

–8

4

y

4

0

2

1 y x 2; x 3, 0, 3 3

y

–4

x

y

4

0

4

1 x 1; x 2, 0, 2, 4 2

y

–4

2

4

8

x

4 –8

–4

0 –4

4

8

x

227


Objective C

To graph a scatter diagram

y Temperature (in °F)

228

39. Chemistry The temperature of a chemical reaction is measured at intervals of 10 min and recorded in the scatter diagram at the right. a. Find the temperature of the reaction after 20 min. b. After how many minutes is the temperature 160F?

300

200 140 0

20

40

60

x

Time (in minutes) The jagged line means that the numbers between 0 and 140 are missing.

Grams of cerium selenate

40. Chemistry The amount of a substance that can be dissolved in a fixed amount of water usually increases as the temperature of the water increases. Cerium selenate, however, does not behave in this manner. The graph at the right shows the number of grams of cerium selenate that will dissolve in 100 mg of water for various temperatures, in degrees Celsius. a. Determine the temperature at which 25 g of cerium selenate will dissolve. b. Determine the number of grams of cerium selenate that will dissolve when the temperature is 80C.

65

85

81

77

89

69

Total Income (in billions of dollars)

2.2

2.6

2.5

2.4

2.7

2.3

2.2

2.6

3.2

2.8

3.5

Average Cost (in dollars)

6.9

6.5

6.3

6.4

6.5

6.1

10 20 40 60 80

Income (in billions of dollars)

3.0

2.5

2.0 65 70 75 80 85 90 Profit (in thousands of dollars)

Average cost (in dollars)

0.7

20

Temperature (in degrees Celsius)

42. Utilities A power company suggests that a larger power plant can produce energy more efficiently and therefore at lower cost to consumers. The table below shows the output and average cost for power plants of various sizes. Draw a scatter diagram for these data. Output (in millions of watts)

30

0

41. Business Past experience of executives of a car company shows that the profit of a dealership will depend on the total income of all the residents of the town in which the dealership is located. The table below shows the profits of several dealerships and the total incomes of the towns. Draw a scatter diagram for these data. Profit (in thousands of dollars)

40

7.0

6.5

6.0 0.5

1.5

2.5

3.5

Output (in millions of watts)

43.

Graph the ordered pairs (x, x2), where x 2, 1, 0, 1, 2.

x 2, 1,

y

−4

−2

1 1 1 1 , , , , 2 3 3 2

y

4

4

2

2

0

, where 1, 2.

44. Graph the ordered pairs x,

2

4

x

−4

−2

0

−2

−2

−4

−4

2

4

x

1 x



229

Section 4.2 / Introduction to Functions

4.2 Objective A

Introduction to Functions To evaluate a function

VIDEO & DVD

CD TUTOR

WEB

SSM

In mathematics and its applications, there are many times when it is necessary to investigate a relationship between two quantities. Here is a financial application: Consider a person who is planning to finance the purchase of a car. If the current interest rate for a 5-year loan is 5%, the equation that describes the relationship between the amount that is borrowed B and the monthly payment P is P 0.018871B.

' (6000, (7000, (8000, (9000,

A relationship between two quantities is not always given by an equation. The table at the right describes a grading scale that defines a relationship between a score on a test and a letter grade. For each score, the table assigns only one letter grade. The ordered pair 84, B indicates that a score of 84 receives a letter grade of B.

113.23) 132.10) 150.97) 169.84)

Score

Grade

90–100 80–89 70–79 60–69 0–59

A B C D F

y

Viscosity

The graph at the right also shows a relationship between two quantities. It is a graph of the viscosity V of SAE 40 motor oil at various temperatures T. Ordered pairs can be approximated from the graph. The ordered pair (120, 250) indicates that the viscosity of the oil at 120ºF is 250 units.

'

0.018871B P

For each amount the purchaser may borrow (B), there is a certain monthly payment (P). The relationship between the amount borrowed and the payment can be recorded as ordered pairs, where the first coordinate is the amount borrowed and the second coordinate is the monthly payment. Some of these ordered pairs are shown at the right.

700 600 500 400 300 200 100 0

(120, 250)

100 120 140

x


Temperature (in °F)

In each of these examples, there is a rule (an equation, a table, or a graph) that determines a certain set of ordered pairs.

Definition of Relation

A relation is a set of ordered pairs.

Here are some of the ordered pairs for the relations given above. Relation Car Payment Grading Scale Oil Viscosity

Some of the Ordered Pairs of the Relation (7500, 141.53), (8750, 165.12), (9390, 177.20) (78, C), (98, A), (70, C), (81, B), (94, A) (100, 500), (120, 250), (130, 200), (150, 180)


Each of these three relations is actually a special type of relation called a function. Functions play an important role in mathematics and its applications.

Definition of Function

A function is a relation in which no two ordered pairs have the same first coordinate and different second coordinates.

The domain of a function is the set of the first coordinates of all the ordered pairs of the function. The range is the set of the second coordinates of all the ordered pairs of the function. For the function defined by the ordered pairs

2, 3, 4, 5, 6, 7, 8, 9 the domain is 2, 4, 6, 8 and the range is 3, 5, 7, 9. Find the domain and range of the function HOW TO

2, 3, 4, 6, 6, 8, 10, 6. The domain is 2, 4, 6, 10.

• The domain of the function is the

The range is 3, 6, 8.

set of the first coordinates of the ordered pairs. • The range of the function is the set of the second coordinates of the ordered pairs.

For each element in the domain of a function there is a corresponding element in the range of the function. A possible diagram for the function above is

Domain

Range

2 4

3 6 8

6 10

{(2, 3), (4, 6), (6, 8), (10, 6)}

Functions defined by tables or graphs, such as those described at the beginning of this section, have important applications. However, a major focus of this text is functions defined by equations in two variables. The square function, which pairs each real number with its square, can be defined by the equation y x2 This equation states that for a given value of x in the domain, the corresponding value of y in the range is the square of x. For instance, if x 6, then y 36 and if x 7, then y 49. Because the value of y depends on the value of x, y is called the dependent variable and x is called the independent variable.


230

231


TA K E N O T E A pictorial representation of the square function is shown at the right. The function acts as a machine that changes a number from the domain into the square of the number.

A function can be thought of as a rule that pairs one number with another number. For instance, the square function pairs a number with its square. The ordered pairs for the values shown at the right are 5, 25,

3

−5 5 0 3

, 0, 0, 3 9 , 5 25

Square

and 3, 9. For this function, the second coordinate is the square of the first coordinate. If we let x represent the first coordinate, then the second coordinate is x2 and we have the ordered pair x, x2.

f(x) = x2

A function cannot have two ordered pairs with different second coordinates and the same first coordinate. However, a function may contain ordered pairs with the same second coordinate. For instance, the square function has the ordered pairs 3, 9 and 3, 9; the second coordinates are the same but the first coordinates are different.

3

−5 5 0 3

The double function pairs a number with twice that number. The ordered pairs for the values shown at the right are 5, 10,

9

25 25 0 9

, 3 6 , 5 5

Double

0, 0, and 3, 6. For this function, the second coordinate is twice the first coordinate. If we let x represent the first coordinate, then the second coordinate is 2x and we have the ordered pair x, 2x.

6

−10 5 0 6

g(x) = 2x

Not every equation in two variables defines a function. For instance, consider the equation y2 x2 9 Because 52 42 9

and

52 42 9


the ordered pairs 4, 5 and 4, 5 are both solutions of the equation. Consequently, there are two ordered pairs that have the same first coordinate 4 but different second coordinates 5 and 5. Therefore, the equation does not define a function. Other ordered pairs for this equation are 0, 3, 0, 3, 7, 4 , and 7, 4 . A graphical representation of these ordered pairs is shown below. Domain 0 4 7

Range −5 −4 −3 3 4 5

Note from this graphical representation that each element from the domain has two arrows pointing to two different elements in the range. Any time this occurs, the situation does not represent a function. However, this diagram does represent a relation. The relation for the values shown is 0, 3, 0, 3, 4, 5, 4, 5, 7, 4 , 7, 4 . The phrase “y is a function of x,” or the same phrase with different variables, is used to describe an equation in two variables that defines a function. To emphasize that the equation represents a function, functional notation is used.

232


Just as the variable x is commonly used to represent a number, the letter f is commonly used to name a function. The square function is written in functional notation as follows: This is the value of the function. It is the number that is paired with x.

b fx x2

l

The name of the function is f.

l

TA K E N O T E The dependent variable y and fx can be used interchangeably.

This is an algebraic expression that defines the relationship between the dependent and independent variables.

The symbol fx is read “the value of f at x” or “f of x.” It is important to note that fx does not mean f times x. The symbol fx is the value of the function and represents the value of the dependent variable for a given value of the independent variable. We often write y fx to emphasize the relationship between the independent variable x and the dependent variable y. Remember that y and fx are different symbols for the same number. The letters used to represent a function are somewhat arbitrary. All of the following equations represent the same function. fx x2 st t2 Pv v2

Each equation represents the square function.

The process of determining fx for a given value of x is called evaluating a function. For instance, to evaluate fx x2 when x 4, replace x by 4 and simplify. fx x2 f4 42 16 The value of the function is 16 when x 4. An ordered pair of the function is 4, 16.

Integrating

Technology See the Projects and Group Activities at the end of this chapter for instructions on using a graphing calculator to evaluate a function. Instructions are also provided in the appendix Keystroke Guide: Evaluating Functions.

Evaluate gt 3t2 5t 1 when t 2.

gt 3t2 5t 1 g2 322 52 1 34 52 1 12 10 1 23

• Replace t by 2 and then simplify.

When t is 2, the value of the function is 23. Therefore, an ordered pair of the function is 2, 23. It is possible to evaluate a function for a variable expression. HOW TO

Evaluate Pz 3z 7 when z 3 h.

Pz 3z 7 P3 h 33 h 7 9 3h 7 3h 2

• Replace z by 3 h and then simplify.

When z is 3 h, the value of the function is 3h 2. Therefore, an ordered pair of the function is 3 h, 3h 2.


HOW TO

233


Recall that the range of a function is found by applying the function to each element of the domain. If the domain contains an infinite number of elements, it may be difficult to find the range. However, if the domain has a finite number of elements, then the range can be found by evaluating the function for each element in the domain.

HOW TO

Find the range of fx x3 x if the domain is 2, 1, 0, 1, 2.

fx x3 x f2 23 2 10 f1 13 1 2 f0 03 0 0 f1 13 1 2 f2 23 2 10

• Replace x by each member of the domain. The range includes the values of f (2), f (1), f (0), f (1), and f (2).

The range is 10, 2, 0, 2, 10. When a function is represented by an equation, the domain of the function is all real numbers for which the value of the function is a real number. For instance: • The domain of fx x2 is all real numbers, because the square of every real number is a real number. • The domain of gx g2

1 22

1 x2

is all real numbers except 2, because when x 2,

1 0

, which is not a real number.

The domain of the grading-scale function is the set of whole numbers from 0 to 100. In set-builder notation, this is written x0 x 100, x whole numbers. The range is A, B, C, D, F.

Score

Grade

90–100 80–89 70–79 60–69 0–59

A B C D F


HOW TO What values, if any, are excluded from the domain of fx 2x2 7x 1? Because the value of 2x2 7x 1 is a real number for any value of x, the domain of the function is all real numbers. No values are excluded from the domain of fx 2x2 7x 1.

Example 1

You Try It 1

Find the domain and range of the function

5, 3, 9, 7, 13, 7, 17, 3.

Find the domain and range of the function

1, 5, 3, 5, 4, 5, 6, 5.

Solution

Your solution

Domain: 5, 9, 13, 17; Range: 3, 7

• The domain is the set of first coordinates. Solution on p. S12

234


Example 2

You Try It 2

Given pr 5r 6r 2, find p3.

Evaluate Gx

Solution

Your solution

3

3x x2

when x 4.

pr 5r 3 6r 2 p3 533 63 2 527 18 2 135 18 2 119

Example 3

You Try It 3

Evaluate Qr 2r 5 when r h 3.

Evaluate fx x2 11 when x 3h.

Solution

Your solution

Qr 2r 5 Qh 3 2h 3 5 2h 6 5 2h 11

Example 4

You Try It 4

Find the range of fx x 1 if the domain is 2, 1, 0, 1, 2. 2

Find the range of hz 3z 1 if the

1 2 3 3

domain is 0, , , 1 .

Solution

To find the range, evaluate the function at each element of the domain.

Your solution

f x x 2 1 f 2 22 1 4 1 3 f 1 12 11 1 0 f 0 02 1 0 1 1 f 1 12 1 1 1 0 f 2 22 1 4 1 3

Example 5

You Try It 5

What is the domain of f x 2x 2 7x 1?

What value is excluded from the domain of f x

2 ? x5

Solution

Because 2x 2 7x 1 evaluates to a real number for any value of x, the domain of the function is all real numbers.

Your solution

Solutions on p. S12


The range is 1, 0, 3. Note that 0 and 3 are listed only once.


235


To evaluate a function

1.

In your own words, explain what a function is.

2.

What is the domain of a function? What is the range of a function?

3.

Does the diagram below represent a function? Explain your answer.

5.

Range

1 2

2 4

−2 −1

9 7

3 4

6 8

0 3

3 0


−3 −1 0 2 4

6.

Domain −4 −2

4 7

1 4

Range

3

1 2 3 4 5

8.

Range


−2 3

Domain

9 12

Domain

Range


6



Domain

Domain

7.

4.

Range

20

Does the diagram below represent a function? Explain your answer. Domain

Range

3

2 4 6 8

For Exercises 9 to 16, state whether the relation is a function. 9. (0, 0), (2, 4), (3, 6), (4, 8), (5, 10)

10. (1, 3), (3, 5), (5, 7), (7, 9)

11.

(2, 1), (4, 5), (0, 1), (3, 5)

12. (3, 1), (1, 1), (0, 1), (2, 6)

13.

(2, 3), (1, 3), (0, 3), (1, 3), (2, 3)

14. (0, 0), (1, 0), (2, 0), (3, 0), (4, 0)

15.

(1, 1), (4, 2), (9, 3), (1, 1), (4, 2)

16. (3, 1), (3, 2), (3, 3), (3, 4)

236

17.

18.


Shipping The table at the right shows the cost to send an overnight package using United Parcel Service. a. Does this table define a function? b. Given x 2.75 lb, find y.

Shipping The table at the right shows the cost to send an “Express Mail” package using the U.S. Postal Service. a. Does this table define a function? b. Given x 0.5 lb, find y.

Weight in pounds (x)

Cost (y)

0<x≤1

$28.25

1<x≤2

$31.25

2<x≤3

$34.75

3<x≤4

$37.75

4<x≤5

$40.75

Weight in pounds (x)

Cost (y)

0 < x ≤ 0.5

$13.65

0.5 < x ≤ 2

$17.85

2<x≤3

$21.05

3<x≤4

$24.20

4<x≤5

$27.30

For Exercises 19 to 22, given f(x) 5x 4, evaluate: 19.

f(3)

20.

f(2)

21.

f(0)

22.

f(1)

For Exercises 23 to 26, given G(t) 4 3t, evaluate: 23.

G(0)

24.

G(3)

25. G(2)

26. G(4)

29. q(2)

30. q(5)

33. F(3)

34. F(6)

37. H(t)

38. H(v)

41. s(a)

42. s(w)

For Exercises 27 to 30, given q(r) r 2 4, evaluate: 27.

q(3)

28.

q(4)

For Exercises 31 to 34, given F(x) x2 3x 4, evaluate: F(4)

32.

F(4)

For Exercises 35 to 38, given H( p) 35.

H(1)

36.

3p , p2

evaluate:

H(3)

For Exercises 39 to 42, given s(t) t3 3t 4, evaluate: 39.

s(1)

40.

s(2)

43. Given P(x) 4x 7, write P(2 h) P(2) in simplest form.

44. Given G(t) 9 2t, write G(3 h) G(3) in simplest form.


31.


45. Business Game Engineering has just completed the programming and testing for a new computer game. The cost to manufacture and package the game depends on the number of units Game Engineering plans to sell. The table at the right shows the cost per game for packaging various quantities. a. Evaluate this function when x 7000. b. Evaluate this function when x 20,000.

46. Airports Airport administrators have a tendency to price airport parking at a rate that discourages people from using the parking lot for long periods of time. The parking rate structure for an airport is given in the table at the right. a. Evaluate this function when t 2.5 h. b. Evaluate this function when t 7 h.

47. Real Estate A real estate appraiser charges a fee that depends on the estimated value, V, of the property. A table giving the fees charged for various estimated values of the real estate appears at the right. a. Evaluate this function when V $5,000,000. b. Evaluate this function when V $767,000.


48.

Shipping The cost to mail a priority overnight package by Federal Express depends on the weight, w, of the package. A table of the costs for selected weights is given at the right. a. Evaluate this function when w 2 lb 3 oz. b. Evaluate this function when w 1.9 lb.

Number of Games Manufactured

237

Cost to Manufacture One Game

0 < x ≤ 2500

$6.00

2500 < x ≤ 5000

$5.50

5000 < x ≤ 10,000

$4.75

10,000 < x ≤ 20,000

$4.00

20,000 < x ≤ 40,000

$3.00

Hours Parked

Cost

00

a0

A point at which a graph crosses the x-axis is called an x-intercept of the graph. The x-intercepts occur when y 0. [11.1B, p. 612]

x

Axis of Symmetry a 4: ≥ 5: < 6: ≤

Plot1 Plot2 Plot3 \Y 1 = X–1< 4 \Y 2 = \Y 3 = \Y 4 = \Y 5 = \Y 6 = \Y 7 =

−10

10

−10

Appendix

Trace

10

Once a graph is drawn, pressing TRACE will place a cursor on the screen, and the coordinates of the point below the cursor are shown at the bottom of the screen. Use the left and right arrow keys to move the cursor along the graph. For the graph at the right, we have f(4.8) 5 3.4592, where f(x) 5 0.1x3 2 2x 1 2 is shown at the top left of the screen.

Y1=.1X^3–2X+2

−10

10

X=4.8

Y=3.4592 −10

In TRACE mode, you can evaluate a function at any value of the independent variable that is within Xmin and Xmax. To do this, first graph the function. Now press TRACE (the value of x) . For the graph at the left below, we used x 5 23.5. If a value of x is chosen outside the window, an error message is displayed. ENTER

10

10

10

Y1=.1X^3–2X+2

Y1=.1X^3–2X+2

Y1=.1X^3–2X+2

−10

10

X=−3.5

−10

10

X=−3.5

−10

Y=4.7125

−10

ERR:INVALID 1: Quit 2: Goto 10

X=55

−10

−10

In the example above where we entered 23.5 for x, the value of the function was calculated as 4.7125. This means that f(23.5) 5 4.7125. The keystrokes 2nd QUIT VARS 11 MATH 1 will convert the decimal value to a fraction.

Y1 Frac 377/80

ENTER

When the TRACE feature is used with two or more graphs, the up and down arrow keys are used to move between the graphs. The graphs below are for the functions f(x) 5 0.1x3 2 2x 1 2 and g(x) 5 2x 2 3. By using the up and down arrows, we can place the cursor on either graph. The right and left arrows are used to move along the graph. 10

10

Y2=2X–3

Y1=.1X^3–2X+2

−10

10

X=−1.4

Y=4.5256

−10

10

X=−1.4

−10

Window

The viewing window for a graph is controlled by pressing WINDOW . Xmin and Xmax are the minimum value and maximum value, respectively, of the independent variable shown on the graph. Xscl is the distance between tic marks on the x-axis. Ymin and Ymax are the minimum value and maximum value, respectively, of the dependent variable shown on the graph. Yscl is the distance between tic marks on the y-axis. Leave Xres as 1.

Y=−5.8 −10

Ymax Yscl Xscl Xmin

Xmax

Ymin

Note: In the standard viewing window, the distance between tic marks on the x-axis is different from the distance between tic marks on the y-axis. This will distort a graph. A more accurate picture of a graph can be created by using a square viewing window. See ZOOM.


764

Appendix A: Keystroke Guide for the TI-83 and TI-83 Plus/TI-84 Plus

Y=

The Y = editor is used to enter the expression for a function. There are ten possible functions, labeled Y1 to Y0, that can be active at any one time. For instance, to enter f(x) 5 x2 1 3x 2 2 as Y1, use the following keystrokes. Y=

X,T,θ X,T, X,T,θ, θ, n

x2

3

X,T,θ X,T, X,T,θ, θ,n

–

765

Plot1 Plot2 Plot3 \Y 1 = X2+3X–2 \Y 2 = \Y 3 = \Y 4 = \Y 5 = \Y 6 = \Y 7 =

2

Note: If an expression is already entered for Y1, place the cursor anywhere on that expression and press CLEAR . Plot1 Plot2 Plot3 \Y 1 = X2+3X– 2 \Y 2 = (2X–1)/(X^3–3) \Y 3 = \Y 4 = \Y 5 = \Y 6 = \Y 7 =

2v 2 1 v3 2 3

To enter s 5 into Y2, place the cursor to the right X,T,θ X,T, θ, n – of the equals sign for Y2. Then press 2 X,T,θ, X,T,θ X,T, X,T,θ, θ,n ^ 1 3 – 3 .

Note: When we enter an equation, the independent variable, v in the expresX,T,θ X,T, θ,n . sion above, is entered using X,T,θ, The dependent variable, s in the expression above, is one of Y1 to Y0. Also note the use of parentheses to ensure the correct order of operations. Observe the black rectangle that covers the equals sign Plot1 Plot2 Plot3 \Y = X +3X–2 for the two examples we have shown. This rectangle \Y = (2X–1)/(X^3–3) means that the function is “active.” If we were to press \Y = \Y = GRAPH , then the graph of both functions would appear. \Y = \Y = You can make a function inactive by using the arrow \Y = keys to move the cursor over the equals sign of that function and then pressing . This will remove the black rectangle. We have done that for Y2, as shown at the right. Now if GRAPH is pressed, only Y1 will be graphed. 1

2

2 3

4

5 6 7

ENTER

It is also possible to control the appearance of the graph by moving the cursor on the Y = screen to the left of any Y. With the cursor in this position, pressing will change the appearance of the graph. The options are shown at the right.

Plot1 Plot2 Plot3 \Y 1 = Default graph line \ Y2 = Bold graph line Y3 = Shade above graph Y 4 = Shade below graph -0Y 5 = Draw path of graph 0Y 6 = Travel path of graph Y 7 = Dashed graph line

ENTER

Zero

The ZERO feature of a graphing calculator is used for various calculations: to find the x-intercepts of a function, to solve some equations, and to find the zero of a function.


x-intercepts To illustrate the procedure for finding x-intercepts, we will use f(x) 5 x2 1 x 2 2. First, use the Y-editor to enter the expression for the function and then graph the function in the standard viewing window. (It may be necessary to adjust this window so that the intercepts are visible.) Once the graph is displayed, use the keystrokes below to find the x-intercepts of the graph of the function. Press .

2nd

CALC (scroll to 2 for zero of the function)

CALCULATE 1 : value 2: zero 3: minimum 4: maximum 5: intersect 6: dy/dx 7: ∫f(x)dx

ENTER

Alternatively, you can just press

2nd

CALC 2.

Left Bound? shown at the bottom of the screen asks you

10

to use the left or right arrow key to move the cursor to the left of the desired x-intercept. Press .

Y1=X^2+X–2

ENTER

−10

10

Left Bound? X=−2.553191

Y=1.9655953

−10

766

Appendix

Right Bound? shown at the bottom of the screen asks you

to use the left or right arrow key to move the cursor to the right of the desired x-intercept. Press .

10

Y1=X^2+X–2

ENTER

−10

10

Right Bound? X=−1.06383 Y=−1.932096 −10

Guess? shown at the bottom of the screen asks you

10

Y1=X^2+X–2

to use the left or right arrow key to move the cursor to the approximate location of the desired x-intercept. Press .

−10

ENTER

10

Guess? X=−2.12766

Y=.39927569 −10

The x-coordinate of an x-intercept is 22. Therefore, an x-intercept is (22, 0).

10

−10

To find the other x-intercept, follow the same steps as above. The screens for this calculation are shown below. 10

10

Y1=X^2+X–2

10

Left Bound? X=.63829787 Y=−.954278 −10

−10

Zero X=−2

Y=0 −10

10

10

Y1=X^2+X–2

−10

10

Y1=X^2+X–2

10

−10

Right Bound? X=1.4893617 Y=1.70756

10

Guess? X=1.0638298 Y=.19556361

−10

−10

10

Zero X=1

Y=0 −10

−10

A second x-intercept is (1, 0). Solve an equation To use the ZERO feature to solve an equation, first rewrite the equation with all terms on one side. For instance, one way to solve the equation x3 2 x 1 1 5 22x 1 3 is first to rewrite it as x3 1 x 2 2 5 0. Enter x3 1 x 2 2 into Y1 and then follow the steps for finding x-intercepts. Find the real zeros of a function the steps for finding x-intercepts.

Pressing ZOOM allows you to select some preset viewing windows. This key also gives you access to ZBox, Zoom In, and Zoom Out. These functions enable you to redraw a selected portion of a graph in a new window. Some windows used frequently in this text are shown below. ZOOM MEMORY 1 : ZBox 2: Zoom In WINDOW 3: Zoom Out Xmin = −4.7 4: ZDecimal Xmax = 4.7 5: ZSquare Xscl = 1 6: ZStandard Ymin = −3.1 7 ZTrig Ymax = 3.1 Yscl = 1 Xres = 1

ZOOM MEMORY 1 : ZBox 2: Zoom In WINDOW 3: Zoom Out Xmin = −15.16129… 4: ZDecimal Xmax = 15.161290… 5: ZSquare Xscl = 1 6: ZStandard Ymin = −10 7 ZTrig Ymax = 10 Yscl = 1 Xres = 1

ZOOM MEMORY 1 : ZBox 2: Zoom In WINDOW 3: Zoom Out Xmin = −10 4: ZDecimal Xmax = 10 5: ZSquare Xscl = 1 6: ZStandard Ymin = −10 7 ZTrig Ymax = 10 Yscl = 1 Xres = 1

ZOOM MEMORY 4 ZDecimal 5: ZSquare WINDOW 6: ZStandard Xmin = −47 7: ZTrig Xmax = 47 8: ZInteger Xscl = 10 9: ZoomStat Ymin = −31 0: ZoomFit Ymax = 31 Yscl = 10 Xres = 1


Zoom

To find the real zeros of a function, follow

Appendix B Proofs and Tables Proofs of Logarithmic Properties In each of the following proofs of logarithmic properties, it is assumed that the Properties of Exponents are true for all real number exponents. The Logarithm Property of the Product of Two Numbers For any positive real numbers x, y, and b, b 1, log b xy log b x log b y. Proof: Let log b x m and log b y n . Write each equation in its equivalent exponential form. Use substitution and the Properties of Exponents. Write the equation in its equivalent logarithmic form. Substitute log b x for m and log b y for n. The Logarithm Property of the Quotient of Two Numbers x For any positive real numbers x, y, and b, b 1, log b log b x log b y. y Proof: Let log b x m and log b y n . Write each equation in its equivalent exponential form. Use substitution and the Properties of Exponents.

Write the equation in its equivalent logarithmic form. Substitute log b x for m and log b y for n.


The Logarithm Property of the Power of a Number For any real numbers x, r, and b, b 1, log b x r r log b x. Proof: Let log b x m . Write the equation in its equivalent exponential form. Raise both sides to the r power. Write the equation in its equivalent logarithmic form. Substitute log b x for m.

x bm y bn m n xy b b xy b m n logb xy m n log b xy log b x log b y

x bm y bn m x b n y b x bmn y x log b m n y x log b log b x log b y y

x bm x r b m r x r b mr log b x r m r log b x r r log b x

767

768

Appendix

Table of Symbols add subtract , , ab multiply a , b

divide

a, b

parentheses, a grouping symbol brackets, a grouping symbol

a

22 pi, a number approximately equal to 7

,

or 3.14 the opposite, or additive inverse, of a

a 1 a

a

the reciprocal, or multiplicative inverse, of a is equal to is approximately equal to is not equal to

/

is less than is less than or equal to is greater than is greater than or equal to an ordered pair whose first component is a and whose second component is b degree (for angles) the principal square root of a the empty set the absolute value of a union of two sets intersection of two sets is an element of (for sets) is not an element of (for sets)

Table of Measurement Abbreviations U.S. Customary System Length

Capacity

Weight

Area

in. ft yd mi

oz c qt gal

oz lb

in2 ft2

inches feet yards miles

fluid ounces cups quarts gallons

ounces pounds

square inches square feet

Metric System Capacity

Weight/Mass

Area

mm

millimeter (0.001 m)

ml

milliliter (0.001 L)

mg

milligram (0.001 g)

cm2

cm dm m dam hm km

centimeter (0.01 m) decimeter (0.1 m) meter decameter (10 m) hectometer (100 m) kilometer (1000 m)

cl dl L dal hl kl

centiliter (0.01 L) deciliter (0.1 L) liter decaliter (10 L) hectoliter (100 L) kiloliter (1000 L)

cg dg g dag hg kg

centigram (0.01 g) decigram (0.1 g) gram decagram (10 g) hectogram (100 g) kilogram (1000 g)

m2

s

seconds

Time h

hours

min minutes

square centimeters square meters Copyright © Houghton Mifflin Company. All rights reserved.

Length

769

Appendix B: Proofs and Tables

Table of Properties Properties of Real Numbers The Associative Property of Addition If a, b, and c are real numbers, then a b c a b c.

The Associative Property of Multiplication If a, b, and c are real numbers, then a b c a b c.

The Commutative Property of Addition If a and b are real numbers, then a b b a.

The Commutative Property of Multiplication If a and b are real numbers, then a b b a.

The Addition Property of Zero If a is a real number, then a 0 0 a a.

The Multiplication Property of One If a is a real number, then a 1 1 a a.

The Multiplication Property of Zero If a is a real number, then a 0 0 a 0.

The Inverse Property of Multiplication If a is a real number and a 0, then

The Inverse Property of Addition If a is a real number, then a a a a 0.

Distributive Property If a, b, and c are real numbers, then ab c ab ac.

a

1 a

1 a

a 1.

Properties of Equations Addition Property of Equations If a b, then a c b c.

Multiplication Property of Equations If a b and c 0, then a c b c.

Properties of Inequalities Addition Property of Inequalities If a b, then a c b c. If a b, then a c b c.

Multiplication Property of Inequalities If a b and c 0, then ac bc. If a b and c 0, then ac bc. If a b and c 0, then ac bc. If a b and c 0, then ac bc.

Properties of Exponents If m and n are integers, then x m x n xmn. If m and n are integers, then x mn x m n.

If m, n, and p are integers, then x m y n p x m p y n p. If n is a positive integer and x 0, then

If x 0, then x0 1.

xn

If m and n are integers and x 0, then

xm xn

xmn.

1 xn

and

1 xn

xn.

If m, n, and p are integers and y 0, then

xm yn

p

x mp . y np


Principle of Zero Products If a b 0, then a 0 or b 0.

Properties of Radical Expressions If a and b are positive real numbers, then ab ab.

If a and b are positive real numbers, then

a a . b b

Property of Squaring Both Sides of an Equation If a and b are real numbers and a b, then a2 b2.

Properties of Logarithms If x, y, and b are positive real numbers and b 1, then logbxy logb x logb y. If x, y, and b are positive real numbers and b 1, then x y

logb logb x logb y.

If x and b are positive real numbers, b 1, and r is any real number, then logb x r r logb x. If x and b are positive real numbers and b 1, then logb b x x.

770

Appendix

Table of Algebraic and Geometric Formulas Slope of a Line

Point-Slope Formula for a Line

Quadratic Formula

y2 y1 m , x1 x 2 x2 x1

y y1 mx x1

x

discriminant b2 4ac

Perimeter and Area of a Triangle, and Sum of the Measures of the Angles

Pythagorean Theorem

Pabc

B a

c

b b2 4ac 2a

A

h

A

C

b

1 bh 2

A B C 180

Perimeter and Area of a Rectangle

W

c

a

b

Perimeter and Area of a Square

P 2L 2W A LW

L

a2 b2 c2

s

P 4s A s2

s

Area of a Trapezoid

Circumference and Area of a Circle

b1 A

h

r

1 hb1 b2 2

C 2 r A r2

b2

L

H

V LWH SA 2LW 2LH 2WH

Volume and Surface Area of a Sphere

r

V

4 r3 3

SA 4 r 2

W

Volume and Surface Area of a Right Circular Cylinder

h r

Volume and Surface Area of a Right Circular Cone

V r 2h SA 2 r 2 2 rh

h r

l

V

1 r 2h 3

SA r 2 rl


Volume and Surface Area of a Rectangular Solid

Solutions to Chapter 1 “You Try It” SECTION 1.1 You Try It 1

You Try It 12

Replace y by each of the elements of the set and determine whether the inequality is true. y 1 5 1 False 1 1 False 5 1 True

z 11 11 0 0 8 8

You Try It 3

100 (43) 57

You Try It 4

(51) 42 17 (102) 9 17 (102) 8 (102) 94

You Try It 5

• Add the seven temperature readings. Solution

Replace z by each element of the set and determine the value of the expression. z (11) 11 (0) 0 (8) 8

6 (7) 0 (5) (8) (1) (1) 28 28 7 4 The average daily low temperature was 4°C.

SECTION 1.2 You Try It 1

You Try It 2

19 (32) 19 32 51 You Try It 3

You Try It 6


You Try It 7

9 (12) 17 4 9 12 (17) (4) 3 (17) (4) 14 (4) 18 8(9)10 72(10) 720

You Try It 8

(2)3(8)7 6(8)7 48(7) 336

You Try It 9

(135) (9) 15

You Try It 10

72 18 4

You Try It 11

36 3 12

To find the average daily low temperature:

• Divide the sum by 7.

The inequality is true for 5. You Try It 2

Strategy

0.444 9 4.000 3 6 40 36 40 36 4

4 0.4 9

125 1 5 100 100 4 125% 125(0.01) 1.25 125% 125

1 1 (100%) 3 3 1 100 % 33 % 3 3

You Try It 4

0.043 0.043(100%) 4.3%

You Try It 5


7 5 3 21 20 18 8 6 4 24 24 24 21 20 18 24 24 24 21 20 18 24 23 23 24 24 You Try It 6

16.127 67.91 16.127 (67.91) 51.783

3

S1

S2

Chapter 1

SECTION 1.3

You Try It 7

The quotient is positive. 3 5 3 12 5 3 8 12 8 12 8 5 3 12 85

1

1

1

1

You Try It 1

18 5[8 2(2 5)] 10 18 5[8 2(3)] 10 18 5[8 6] 10 18 5[14] 10 18 70 10 18 7 11

You Try It 2

36 (8 5)2 (3)2 2 36 (3)2 (3)2 2 36 9 9 2 492 4 18 14

You Try It 3

(6.97 4.72)2 4.5 0.05 (2.25)2 4.5 0.05 5.0625 4.5 0.05 22.78125 0.05 455.625

3223 9 2225 10 The product is negative.

You Try It 8

5.44 3.8 4352 16322 20.672 5.44(3.8) 20.672 You Try It 9

63 (6 6 6) 216 You Try It 10

(3)4 (3)(3)(3)(3) 81


You Try It 1

You Try It 12

2 5

2

2 5

2 5

4 25

You Try It 13

3(0.3)3 3(0.3)(0.3)(0.3) 0.9(0.3)(0.3) 0.27(0.3) 0.081

You Try It 2

x3 2(x y) z2 (2)3 2[2 (4)] (3)2 (2)3 2(2) (3)2 8 2(2) 9 849 12 9 21

You Try It 3

3a 2b 5a 6b 2a 4b

You Try It 4

3y2 7 8y2 14 5y2 7

You Try It 5

5(4y2) 20y2

You Try It 6

7(2a) 14a

You Try It 7

(5x)(2) 10x

You Try It 8

8(2a 7b) 16a 56b

You Try It 14

532 516 2 516 2 5 42 202 You Try It 15

216 36 6 36 6 66

You Try It 16 Strategy

Solution

To determine the annual net income for 2005, multiply the net income for the third quarter of 2004 (2.1) by the number of quarters in one year (4). 4(2.1) 8.4 The annual net income for Frontier Airlines for 2005 would be $8.4 million.

a2 b2 ab 52 (3)2 25 9 5 (3) 5 (3) 34 2 17


(33)(2)3 (3)(3)(3) (2)(2)(2) 27(8) 216

S3

Solutions to You Try It

You Try It 9

(3a 1)5 15a 5

You Try It 3

A B {2, 1, 0, 1, 2, 3, 4}

You Try It 10

2(x2 x 7) 2x2 2x 14

You Try It 4

C D {10, 16}

You Try It 11

3y 2(y 7x) 3y 2y 14x 14x y

You Try It 5

AB

You Try It 6

{xx 59, x positive even integers}

2(x 2y) (x 3y) 2x 4y x 3y x y

You Try It 7

{x x 3, x real numbers}

You Try It 13

You Try It 8

The graph is the numbers greater than 2.

You Try It 12

3y 2[x 4(2 3y)] 3y 2[x 8 12y] 3y 2x 16 24y 2x 21y 16 You Try It 14

the unknown number: x the difference between the number and sixty: x 60

−5 − 4 − 3 − 2 −1

You Try It 9

0

1

2

3

4

5

The graph is the numbers greater than 1 and the numbers less than 3.

5(x 60); 5x 300 You Try It 15

You Try It 16

the speed of the older model: s the speed of the new model: 2s

−5 − 4 −3 − 2 − 1

You Try It 10

the length of the longer piece: L the length of the shorter piece: 6L

You Try It 1 You Try It 2

You Try It 11

A {9, 7, 5, 3, 1}

1

2

3

4

5

The graph is the numbers less than or equal to 4 and greater than or equal to 4. −5 −4 −3 − 2 − 1

SECTION 1.5

0

0

1

2

3

4

5

The graph is the real numbers. −5 − 4 −3 − 2 − 1

0

1

2

3

4

5

A {1, 3, 5, . . .}

Solutions to Chapter 2 “You Try It” Copyright © Houghton Mifflin Company. All rights reserved.


You Try It 3

5 4x 8x 2 1 1 54 2 8 4 4 51 22 44

10x x2 3x 10 105 52 35 10 50 25 15 10 25 5 No, 5 is not a solution.

3 8 3 3 8 8

The solution is

1 Yes, is a solution. 4

You Try It 2

5 y 6 5 3 y 6 8 29 y 24

You Try It 4

29 . 24

2x 6 5

5 2

2 5 x 6 5 2 x 15

The solution is 15.

•

2 2x x 5 5

Chapter 2

You Try It 5

4x 8x 16 4x 16 4x 16 4 4 x 4


To find the number of ounces of cereal in the bowl, solve Q Ar for A using Q 2 and r 25% 0.25.

Solution

Q 2 2 0.25 8

The solution is 4. You Try It 6

PBA 1 B 18 6 1 6 B 6 18 6 B 108 18 is 16

• 16

2 1 % 3 6

2 % of 108. 3

Solution

Use the percent equation. B 83.3, the total revenue received by the BCS; A 3.1, the amount received by the college representing the Pac-10 conference; P is the unknown percent.


To find the distance, solve the equation d rt for d. The time is 3 h. Therefore, t 3. The plane is moving against the wind, which means the headwind is slowing the actual speed of the plane. 250 mph 25 mph 225 mph. Thus r 225.

Solution

d rt d 2253 675

PBA

• B 83.3, A 3.1 P83.3 3.1 3.1 P 0.037 83.3

The college representing the Pac-10 conference received approximately 3.7% of the BCS revenue.


Strategy

5x 7 10 5x 7 7 10 7 5x 3 5x 3 5 5 3 x 5 The solution is

• Subtract 11. • Divide by 3.

The solution is 3. 5 2x 5 8 3 4

You Try It 3

Clarissa must invest $800 in the bank account.

3 . 5

9 3x 9 3x 3 3 3 x

The interest earned on the municipal bond was $64. • I 64, r 0.08, t 1

• Divide by 5.

2 11 11 11 3x

Solution

Prt P0.081 0.08P 0.08P 0.08 P

• Subtract 7.

2 11 3x

You Try It 2

I Prt 10000.0641 64

I 64 64 64 0.08 800

• r 225, t 3

The plane travels 675 mi in 3 h.

You Try It 8

To find how much she must deposit into the bank account: • Find the amount of interest earned on the municipal bond by solving I Prt for I using P 1000, r 6.4% 0.064, and t 1. • Solve I Prt for P using the amount of interest earned on the municipal bond as I. r 8% 0.08, and t 1.

• Q 2, r 0.25

The cereal bowl contains 8 oz of cereal.


Ar A0.25 A0.25 0.25 A

5 5 2 5 5 x 8 8 3 4 8

2 5 x 3 8

3 2 x 2 3

3 2

5 8

• Recall that 2x 2 x. 3 3 • Multiply by 3 . 2


S4


x

15 16

The solution is

15 . 16

You Try It 4

6

• Distributive Property • Subtract 18. • Divide by 4.

3 4

The solution is 6.


You Try It 6

5x 4 6 10x 5x 10x 4 6 10x 10x 5x 4 6 5x 4 4 6 4 5x 2 5x 2 5 5 2 x 5

5x 43 2x 23x 2 6 5x 12 8x 6x 4 6 13x 12 6x 2 13x 6x 12 6x 6x 2 7x 12 2 7x 12 12 2 12 7x 14 7x 14 7 7 x2

• Distributive Property • Subtract 6x. • Add 12. • Divide by 7.

You Try It 9

x 5 4x 25 5x 5 25 5x 5 5 25 5 5x 30 5x 30 5 5 x6

2 3x 52x 3 3x 8 2 3x 10x 15 3x 8 2 7x 15 3x 8 14x 30 3x 8 14x 3x 30 3x 3x 8 11x 30 8 11x 30 30 8 30 11x 22 11x 22 11 11 x2

• Distributive Property • Subtract 3x. • Add 30. • Divide by 11.

The solution is 2. • Subtract 10x. • Subtract 4.


Given: F1 45 F2 80 d 25 Unknown: x

Solution

F1 x 45x 45x 45x 80x 125x 125x 125 x

• Divide by 5.

2 5

The solution is . You Try It 7

5x 10 3x 6 4x 2x 10 6 4x 2x 4x 10 6 4x 4x 6x 10 6 6x 10 10 6 10 6x 16 16 6x 6 6

8 . 3

The solution is 2.

The solution is . You Try It 5

The solution is

8 3

You Try It 8

2 7 x3 3 2

2 7 x3 6 3 2 2 7 6 x 63 6 3 2 4x 18 21 4x 18 18 21 18 4x 3 4x 3 4 4 3 x 4

x

S5

• Combine like terms. • Add 4x. • Add 10. • Divide by 6.

F2 d x 8025 x 2000 80x 2000 80x 80x 2000 2000 125 16

The fulcrum is 16 ft from the 45-pound force.

Chapter 2

17.50x 157.50 17.50 17.50 x9


The smaller number: n The larger number: 12 n The total of three times the smaller number and six

You are purchasing 9 tickets.

amounts to

seven less than the product of four and the larger number

3n 6 412 n 7 3n 6 48 4n 7 3n 6 41 4n 3n 4n 6 41 4n 4n 7n 6 41 7n 6 6 41 6 7n 35 7n 35 7 7 n5 12 n 12 5 7 The smaller number is 5. The larger number is 7.

Solution

Strategy

To find the length, write and solve an equation using x to represent the length of the shorter piece and 22 x to represent the length of the longer piece. Solution

The length of the longer piece

is

4 in. more than twice the length of the shorter piece

22 x 2x 4 22 x 2x 2x 2x 4 22 3x 4 22 22 3x 4 22 3x 18 3x 18 3 3 x6 22 x 22 6 16


You Try It 4

• First integer: n Second integer: n 1 Third integer: n 2 • The sum of the three integers is 6. n n 1 n 2 6 3n 3 6 3n 9 n 3 n 1 3 1 2 n 2 3 2 1 The three consecutive integers are 3, 2, and 1.


To find the number of tickets that you are purchasing, write and solve an equation using x to represent the number of tickets purchased.

The length of the shorter piece is 6 in. The length of the longer piece is 16 in.

SECTION 2.4 You Try It 1 Strategy

$.80 fertilizer $.55 fertilizer $.75 fertilizer

Amount

Cost

Value

20

.80

0.80(20)

x

.55

0.55x

20 x

.75

0.7520 x

• The sum of the values before mixing

Solution

$3.50 plus $17.50 for each ticket

• Pounds of $.55 fertilizer: x

equals the value after mixing. is

$161

3.50 17.50x 161 3.50 3.50 17.50x 161 3.50 17.50x 157.50

Solution

0.8020 0.55x 0.7520 x 16 0.55x 15 0.75x 16 0.20x 15 0.20x 1 x5 5 lb of the $.55 fertilizer must be added.


S6


• The distance out equals the

You Try It 2

distance back.

• Liters of 6% solution: x

Strategy

150t 1005 t 150t 500 100t 250t 500 t 2 (The time out was 2 h.)

Solution Amount

Percent

Quantity

x

0.06

0.06x

5

0.12

50.12

x5

0.08

0.08x 5


The distance out 150t 1502 300 mi The parcel of land was 300 mi away.

• The sum of the quantities before mixing equals the quantity after mixing. Solution

S7

0.06x 50.12 0.06x 0.60 0.02x 0.60 0.02x x

0.08x 5 0.08x 0.40 0.40 0.20 10

The pharmacist adds 10 L of the 6% solution to the 12% solution to get an 8% solution.


2x 1 6x 7 4x 1 7 4x 8 4x 8 4 4 x 2 x x 2 −5 −4 −3 −2 −1

0

1

• Subtract 6x from each side. • Add 1 to each side. • Divide each side by 4.

2

3

4

5


• Rate of the first train: r

You Try It 2

Rate of the second train: 2r

1st train 2nd train

Rate

Time

Distance

r 2r

3 3

3r 32r

• The sum of the distances traveled

by the two trains equals 288 mi.


Solution

3r 32r 3r 6r 9r r

288 288 288 32

2 5x 3 13 2 3 5x 3 3 13 3

• Subtract 3 from each of the three parts.

The first train is traveling at 32 mph. The second train is traveling at 64 mph.

5 5x 10 5 5x 10 5 5 5 1 x 2 x1 x 2

• Time spent flying out: t

You Try It 4

You Try It 4

Time spent flying back: 5 t

Out Back

You Try It 3

2r 232 64

Strategy

5x 2 4 3x 2 5x 2 4 3x 6 5x 2 10 3x 8x 2 10 8x 12 8x 12 8 8 3 x 2 3 xx 2

Rate

Time

Distance

150 100

t 5t

150t 1005 t

2 3x 11 3x 9 x 3 x x 3

• Divide each of the three parts by 5.

or 5 2x 7 2x 2 x1 x x 1

x x 3 x x 1 x x 3 or x 1

Chapter 2


x 3 2

You Try It 2

To find the maximum height, substitute the given values in the inequality

There is no solution to this equation because the absolute value of a number must be nonnegative.

1 bh A and solve. 2

1 bh A 2

Solution

5 3x 5 3 3x 5 2 • Subtract 5. 3x 5 2 • Multiply by 1. 3x 5 2 3x 5 2 3x 3 3x 7 7 x 1 x 3

You Try It 3

1 12x 2 50 2 6x 2 50 6x 12 50 6x 38 19 x

3

7 3

The solutions are 1 and .

The largest integer less than

19 is 6. 3

3x 2 8

You Try It 4

8 3x 2 8 8 2 3x 2 2 8 2 10 3x 6 10 3x 6

3 3 3 10

x 2 3

x2628 The maximum height of the triangle is 8 in.


To find the range of scores, write and solve an inequality using N to represent the score on the last test.

x

Solution

72 94 83 70 N 89 5 319 N 80 89 5 319 N 5 80 5 5 89 5 400 319 N 445 400 319 319 N 319 445 319 81 N 126 80

The absolute value of a number must be nonnegative.

The solution set is the empty set.

2x 7 1

You Try It 6

The absolute value of a number is nonnegative. The solution set is the set of real numbers. You Try It 7

5x 3 8


2x 3 5 2x 3 5 2x 8 x4

3x 7 0

You Try It 5

Because 100 is the maximum score, the range of scores to receive a B grade is 81 N 100.

10

x 2 3

2x 3 5 2x 2 x 1


• Add 3. • Divide by 2.

5x 3 8 5x 3 8 or 5x 11 5x 5 11 x x1 5 11 x x 1 x x 5 11 x x x x 1 5 11 x x or x 1 5


S8


You Try It 8

Solution

Strategy

Let b represent the diameter of the bushing, T the tolerance, and d the lower and upper limits of the diameter. Solve the absolute value inequality d b T for d.

S9

d b T d 2.55 0.003 0.003 d 2.55 0.003 0.003 2.55 d 2.55 2.55 0.003 2.55 2.547 d 2.553 The lower and upper limits of the diameter of the bushing are 2.547 in. and 2.553 in.

Solutions to Chapter 3 “You Try It” SECTION 3.1 You Try It 1

You Try It 5

QR RS ST QT 24 RS 17 62 41 RS 62 RS 21 RS 21 cm

You Try It 2

Strategy

The angles labeled are adjacent angles of intersecting lines and are, therefore, supplementary angles. To find x, write an equation and solve for x.

Solution

(x 16°) 3x 180° 4x 16° 180° 4x 164° x 41°

AC AB BC 1 AC (BC) BC 4 1 AC (16) 16 4 AC 4 16 AC 20


3x y because corresponding angles have the same measure. y (x 40°) 180° because adjacent angles of intersecting lines are supplementary angles. Substitute 3x for y and solve for x.

Solution

3x (x 40°) 180° 4x 40° 180° 4x 140° x 35°

AC 20 ft You Try It 3


Strategy

Solution

Supplementary angles are two angles whose sum is 180°. To find the supplement, let x represent the supplement of a 129° angle. Write an equation and solve for x. x 129° 180° x 51° The supplement of a 129° angle is a 51° angle.


To find the measure of a, write an equation using the fact that the sum of the measure of a and 68° is 118°. Solve for a.

Solution

a 68° 118° a 50° The measure of a is 50°.


• To find the measure of angle b, use the fact that b and x are supplementary angles. • To find the measure of angle c, use the fact that the sum of the measures of the interior angles of a triangle is 180°. • To find the measure of angle y, use the fact that c and y are vertical angles.

S10

Chapter 3

Solution

b x 180° b 100° 180° b 80°

You Try It 3

a b c 180° 45° 80° c 180° 125° c 180° c 55°

Strategy

To find the circumference, use the circumference formula that involves the diameter. Leave the answer in terms of .

Solution

C d C (9) C 9

y c 55°

The circumference is 9 in. You Try It 8 Strategy

To find the measure of the third angle, use the facts that the measure of a right angle is 90° and the sum of the measures of the interior angles of a triangle is 180°. Write an equation using x to represent the measure of the third angle. Solve the equation for x.

Solution

x 90° 34° 180° x 124° 180° x 56°


To find the number of rolls of wallpaper to be purchased: • Use the formula for the area of a rectangle to find the area of one wall. • Multiply the area of one wall by the number of walls to be covered (2). • Divide the area of wall to be covered by the area that one roll of wallpaper will cover (30).

Solution

A LW A 12 8 96

The measure of the third angle is 56°.

2(96) 192

SECTION 3.2

192 30 6.4


To find the perimeter, use the formula for the perimeter of a square. Substitute 60 for s and solve for P.

Solution

P 4s P 4(60) P 240

Because a portion of a seventh roll is needed, 7 rolls of wallpaper should be purchased.

Strategy

To find the area, use the formula for the area of a circle. An approximation is asked for; use the key on a calculator. r 11

Solution

A r2 A (11)2 A 121 A 380.13

You Try It 2

To find the perimeter, use the formula for the perimeter of a rectangle. 1

The area is approximately 380.13 cm2.

Substitute 11 for L and 8 2 for W and solve for P. Solution

P 2L 2W

1 2 17 P 2(11) 2 2 P 22 17 P 39 P 2(11) 2 8

The perimeter of a standard piece of typing paper is 39 in.


To find the volume, use the formula for the volume of a cube. s 2.5

Solution

V s3 V (2.5)3 15.625 The volume of the cube is 15.625 m3.


You Try It 5

The perimeter of the infield is 240 ft.

Strategy

The area of one wall is 96 ft 2. The area of the two walls is 192 ft 2.



Solution

You Try It 4

To find the volume: • Find the radius of the base of the cylinder. d 8 • Use the formula for the volume of a cylinder. Leave the answer in terms of .

Strategy

To find which solid has the larger surface area: • Use the formula for the surface area of a cube to find the surface area of the cube. s 10 • Find the radius of the sphere. d 8 • Use the formula for the surface area of a sphere to find the surface area of the sphere. Because this number is to be compared to another number, use the key on a calculator to approximate the surface area. • Compare the two numbers.

Solution

SA 6s2 SA 6(10)2 6(100) 600 The surface area of the cube is 600 cm2.

1 1 d (8) 4 2 2 V r 2h (4)2(22) (16)(22) 352 r

The volume of the cylinder is 352 ft 3. You Try It 3 Strategy

Solution

S11

To find the surface area of the cylinder: • Find the radius of the base of the cylinder. d 6 • Use the formula for the surface area of a cylinder. An approximation is asked for; use the key on a calculator.

1 1 d (8) 4 2 2

r

SA 4 r 2 SA 4 (4)2 4 (16) 64 201.06 The surface area of the sphere is approximately 201.06 cm2.

1 1 r d (6) 3 2 2 SA 2 r 2 2 rh SA 2 (3)2 2 (3)(8) SA 2 (9) 2 (3)(8) SA 18 48 SA 66 SA 207.35

600 201.06 The cube has a larger surface area than the sphere.

The surface area of the cylinder is approximately 207.35 ft2.

Solutions to Chapter 4 “You Try It” Copyright © Houghton Mifflin Company. All rights reserved.

Section 4.1

You Try It 3

You Try It 1

2 3(4) 14 2 12 14 14 14

4 2 –4

–2

0

x 3y 14

2

Yes, (2, 4) is a solution of x 3y 14.

4

–2 –4

You Try It 2

The coordinates of A are (4, 2). The coordinates of B are (2, 4). The abscissa of D is 0. The ordinate of C is 0.

You Try It 4

Replace x by 2 and solve for y. y

3(2) 6 3x 6 x1 2 1 1

The ordered-pair solution is (2, 6).

S12

Chapter 4

You Try It 5

1 y x2 2

x

1 (4) 2 2 1 (2) 2 2 1 (0) 2 2 1 (2) 2 2

4 2 0 2

4

(4, 4)

For x 5, 5 5 0.

3

(2, 3)

f 5

2

(0, 2)

number.

1

(2, 1)

5 is excluded from the domain of the function.

y

–4

–2

2

2 x5

(x, y)

You Try It 5

2 2 , which is not a real 55 0

SECTION 4.3

(−4, 4) 4 (−2, 3)

f x

y

(0, 2) (2, 1)

0

2

4

You Try It 1

y

x

4 2

–2 –4

–4

–2

0

2

4

2

4

x

–2

You Try It 6

–4

To draw a scatter diagram: • Draw a coordinate grid with the horizontal axis representing the year and the vertical axis representing the number of deaths. • Graph the ordered pairs 1998, 14, 1999, 28, 2000, 17, 2001, 18, (2002, 23) and 2003, 29.

You Try It 2 2 –4

Number of deaths

–2

x

–4

3x 2y 4

y

2y 3x 4 3 y x2 2

y 30 25 20 15 10 5 0

0 –2

You Try It 3 Solution

y 4

4 2 –4

'98 '99 '00 '01 '02 '03

You Try It 4

Domain: 1, 3, 4, 6 Range: 5

goes through the point (0, 3).

• The domain is

2

the set of first coordinates.

3x x2 12 34 G4 6 4 2 2

–4

–2

f x x2 11

f 3h 3h2 11 9h2 11 hz 3z 1 h0 30 1 1 1 1 3 12 h 3 3

2 2 3 13 3 3 h1 31 1 4

0

2

4

x

–2

Gx

h

x

• The graph of y 3

y

You Try It 4

4

–4

y30 y3 4

You Try It 3

2

x

SECTION 4.2

You Try It 2

0 –2

Year

You Try It 1

–2

–4

You Try It 5

x-intercept: 3x y 2 3x 0 2 3x 2 2 x 3 x-intercept:

• Let y 0.

2 ,0 3

y-intercept: 3x y 2 30 y 2 y 2

x 0.

y 2 y-intercept: 0, 2 y 4 2 –4

–2

0 –2 –4

The range is 1, 2, 3, 4 .

• Let

2

4

x


Strategy


55,000 25,000 • x1, y1 5, 25,000, x2, y2 2, 55,000 25 30,000 3 10,000

You Try It 6

x-intercept: 1 y x1 4 1 0 x1 4 1 x1 4 x 4 4, 0

y-intercept: 0, b b1 0, 1

A slope of 10,000 means that the value of the recycling truck is decreasing by $10,000 per year. You Try It 4

y 4 2 –4

–2

0

2

4

x

–2 –4

2 x 3y 6 3y 2 x 6 2 y x2 3 2 2 m 3 3 y-intercept (0, 2) y 4

You Try It 7 Height (in inches)

80

(32, 74)

40 20 10 20 30 40

–4

When L 20, h 65. When L 40, h 80.

60

0

2

• Graph h 34 L 50.

h

–2

0

2

4

x

–2 –4

L

Stride (in inches)

The ordered pair 32, 74 means that a person with a stride of 32 in. is 74 in. tall.

You Try It 5

x1, y1 3, 2 m3 y 4 2 –4

SECTION 4.4

–2

0

2

4

x

–2 –4

You Try It 1

Let P1 4, 3 and P2 2, 7. m

y2 y1 10 7 3 5 x2 x1 24 2

SECTION 4.5

The slope is 5. Copyright © Houghton Mifflin Company. All rights reserved.


Let P1 6, 1 and P2 6, 7. y2 y1 7 1 8 m x2 x1 66 0

m

1 3

x1, y1 3, 2

P1 5, 25,000, P2 2, 55,000

y y1 mx x1 1 y 2 x 3

3 1 y 2 x 3 3 1 y2 x1 3 1 y x3 3

m

The equation of the line is

Division by zero is not defined. The slope of the line is undefined.

You Try It 3

y2 y1 x2 x1

S13

1 3

y x 3.

Chapter 4

You Try It 2

m 3

x1, y1 4, 3

y y1 mx x1 y 3 3x 4 y 3 3x 12 y 3x 9 The equation of the line is y 3x 9. You Try It 3

Let x1, y1 2, 0 and x2, y2 5, 3. 30 3 y2 y1 1 m x2 x1 52 3


1 3 4 7 2 9 5 1 6 m2 3 64 2 4 4 m 1 m 2 3 9 3

You Try It 2

The equation of the line is y x 2.

0 33 y2 y1 0 x2 x1 5 2 7

Strategy

Solution

dependent variables. The function is to be used to predict the Celsius temperature, so that quantity is the dependent variable, y. The Fahrenheit temperature is the independent variable, x. • From the given data, two ordered pairs are (212, 100) and (32, 0). Use these ordered pairs to determine the linear function. Let x1, y1 32, 0 and x2, y2 212, 100.

5 F 32. 9

x 4y 3 4y x 3 3 1 y x 4 4 m1

1 4

m1 m2 1 1 m2 1 4 m2 4 y y1 mx x1 y 2 4 x 2 • x1, y1 2, 2 y 2 4x 2 y 2 4x 8 y 4x 6

m

f F

5 2

Yes, the lines are parallel.

• Select the independent and

The linear function is

5 2

m1 m2

You Try It 3

y2 y1 100 0 100 5 x2 x1 212 32 180 9 y y1 mx x1 5 y 0 x 32 9 5 5 y x 32, or C F 32. 9 9

5x 2y 2 2y 5x 2 5 y x1 2 5 m1 2

m2

The line has zero slope. The line is a horizontal line. All points on the line have an ordinate of 3. The equation of the line is y 3. You Try It 5

x2, y2 6, 5

5x 2y 6 2y 5x 6 5 y x3 2

Let x1, y1 2, 3 and x2, y2 5, 3. m

x2, y2 7, 1

• x1, y1 4, 1,

No, the lines are not perpendicular.

y y1 mx x1 y 0 1x 2 y 1x 2 yx2

You Try It 4

• x1, y1 2, 3,

m1

The equation of the line is y 4x 6.


x 3y 6 3y x 6 1 y x2 3


S14


y

You Try It 2

y 2 y

4 2 –4 –2 0 –2

S15

4 2

x

4

2 –4 –2 0 –2

–4

2

x

4

–4

Solutions to Chapter 5 “You Try It” Substitute into Equation (2).


y 4

of intersection of the graphs of the equations.

2 −4

−2

0

6x 3y 4 6x 33x 3 4 6x 9x 9 4 15x 9 4 15x 5 5 1 x 15 3

• Find the point

2

x

4

−2 −4

The solution is 1, 2. You Try It 2

y

Substitute the value of x into Equation (1).

• Graph the two

4

3x y 3 1 3 y3 3 1y3 y 2 y 2

equations.

2 −4

−2

0

2

x

4

−2 −4

The lines are parallel and therefore do not intersect. The system of equations has no solution. The system of equations is inconsistent. You Try It 3

y

• Graph the two

4

equations.

2 −4

−2

0

2

x

4


−2 −4

The two equations represent the same line. The system of equations is dependent. The solutions are the

ordered pairs x, You Try It 4

(1) (2)

3 x3 . 4

3x y 3 6x 3y 4

The solution is You Try It 5

1 , 2 . 3

y 2x 3 3 x 2y 6 3 x 2y 6 3 x 22 x 3 6 3 x 4x 6 6 x 6 6 x 0 x0 (1) (2)

Substitute the value of x into Equation (1). y 2x 3 y 20 3 y03 y 3 The solution is 0, 3. y

• Graph the two

4

Solve Equation (1) for y. 3x y 3 y 3x 3 y 3x 3

equations.

2 –4

–2

0

2

–2 –4

(0, −3)

4

x

Chapter 5

You Try It 6

6x 3y 6 2x y 2

(1) (2)


Solve Equation (2) for y. 2x y 2 y 2 x 2 y 2x 2

Write Equation (2) in the form Ax By C. 3x 2y 6x 2 3x 2y 2

Substitute into Equation (1). 6x 3y 6x 32 x 2 6x 6x 6 6

6 6 6 6

Solve the system:

4x 10 y 12 15x 10 y 10 11x 22 x 2

• Graph the two

4

equations.

2 −4

−2

0

2

4

x

2 x 5y 22 5y 4 5y 5y y

−4

You Try It 7

• Amount invested at 4.2%: x Amount invested at 6%: y

Amount at 4.2% Amount at 6%

Principal

Rate

Interest

x

0.042

0.042x

y

0.06

0.06y

You Try It 2

2x y 5 4x 2y 6 Eliminate y. 22 x y 25 4x 2y 6 4x 2y 10 4x 2y 6 0x 0y 4 0 4

The two accounts earn the same interest. x y 13,600 0.042 x 0.06y 10 x y 7 10 Substitute y for x in 7

x y 13,600 and solve for y. 10 y y 13,600 7 17 y 13,600 7 y 5600 x 5600 13,600 x 8000 $8000 must be invested at 4.2% and $5600 must be invested at 6%.

6 6 6 10 2

The solution is 2, 2.

• The total investment is $13,600.

Solution

• Add the equations. • Solve for x.

Replace x in Equation (1).

−2

Strategy

2 x 5y 6 3x 2y 2

Eliminate y. 22 x 5y 26 53x 2y 52

The system of equations is dependent. The solutions are the ordered pairs x, 2 x 2. y

(1) 2 x 5y 6 (2) 3x 2y 6x 2

• Add the equations.

This is not a true equation. The system is inconsistent and therefore has no solution. You Try It 3

(1) (2) (3)

xyz6 2 x 3y z 1 x 2y 2z 5

Eliminate z. Add Equations (1) and (2). xyz6 2 x 3y z 1 3x 2y 7

• Equation (4)

Multiply Equation (2) by 2 and add to Equation (3). 4x 6y 2z 2 x 2y 2z 5 5x 8y 7 • Equation (5)


S16

S17


Solve the system of two equations. (4) 3x 2y 7 (5) 5x 8y 7


• Cost of an orange tree: x Cost of a grapefruit tree: y

Multiply Equation (4) by 4 and add to Equation (5). 12 x 8y 28 5x 8y 7 7x 21 x3 Replace x by 3 in Equation (4). 3x 2y 7 33 2y 7 9 2y 7 2y 2 y 1

First purchase: Unit Cost

Value

25 20

x y

25x 20y

Amount

Unit Cost

Value

20 30

x y

20x 30y

Orange trees Grapefruit trees

Second purchase:

Orange trees Grapefruit trees

Replace x by 3 and y by 1 in Equation (1).

• The total of the first purchase was $290. The total of the second purchase was $330.

xyz6 3 1 z 6 4z6 z2

Solution

25x 20y 290

425x 20y 4 290

20x 30y 330

520x 30y 5 330

• Multiply by 4.

The solution is 3, 1, 2.

• Multiply by 5.

100x 80y 1160 100x 150y 1650 70y 490 y7


• Rate of the rowing team in calm water: t Rate of the current: c With current Against current

Amount

Rate

Time

Distance

tc tc

2 2

2t c 2t c

25x 20y 290 25x 207 290 25x 140 290 25x 150 x6

• y7

The cost of an orange tree is $6. The cost of a grapefruit tree is $7.


• The distance traveled with the current is 18 mi. The distance traveled against the current is 10 mi.

SECTION 5.4

2t c 18 2t c 10

You Try It 1

Solution 2t c 18

2t c 10

tc9 7c9 c2

1 1 2t c 18 2 2 1 1 2t c 10 2 2 tc9 tc5 2t 14 t7

Shade above the solid line y 2 x 3. Shade above the dotted line y 3x. The solution set of the system is the intersection of the solution sets of the individual inequalities. y 4 2 −4

• Substitute 7 for t.

The rate of the rowing team in calm water is 7 mph. The rate of the current is 2 mph.

−2

0 −2 −4

2

4

x

S18

Chapter 6

You Try It 2

3x 4y 12

The solution set of the system is the intersection of the solution sets of the individual inequalities.

4y 3x 12 y

3 x3 4

y

Shade above the dotted line

4

3 4

2

y x 3. −4

−2

0

Shade below the dotted line

−2

3 y x 1. 4

−4

2

4

x

Solutions to Chapter 6 “You Try It” SECTION 6.1


You Try It 1

3a2b42ab34 3a2b4 24a4b12

3a2b416a4b12 48a6b16 yn32 yn32 • Multiply the exponents. y2n6

You Try It 3

ab33 4 a3b9 4 a12b36

You Try It 4

You Try It 5

You Try It 6

Solution

4 5r23s2t5 20r2t5 3 2 16r s 44 5rs2 5 4t 9u6v41 91u6v4 3 2 2 6u v 62u6v4 91 62u0v8 36 8 9v 4 8 v a2n1 a2n1n3 an3 a2n1n3 an2

• Subtract the exponents.

You Try It 7

942,000,000 9.42 108

You Try It 8

2.7 105 0.000027

You Try It 9

5,600,000 0.000000081 900 0.000000028 5.6 106 8.1 108 9 102 2.8 108 5.68.1 106828 92.8 1.8 104 18,000

1 107 1 107 60 6 10 6 10 107 6 108 The computer can perform 6 108 operations in 1 min.


Rx 2 x4 5x3 2 x 8 R2 224 523 22 8 216 58 4 8 32 40 4 8 76

• Replace x by 2. Simplify.

You Try It 2

The leading coefficient is 3, the constant term is 12, and the degree is 4.

You Try It 3

a. Yes, this is a polynomial function. b. No, this is not a polynomial function. A polynomial function does not have a variable expression raised to a negative power. c. No, this is not a polynomial function. A polynomial function does not have a variable expression within a radical.


You Try It 2

To find the number of arithmetic operations: • Find the reciprocal of 1 107, which is the number of operations performed in 1 s. • Write the number of seconds in 1 min (60) in scientific notation. • Multiply the number of arithmetic operations per second by the number of seconds in 1 min.


x

You Try It 4

y

4 3 2 1 0 1 2

y

5 0 3 4 3 0 5

You Try It 3

4 2 –4

–2

0

S19

2

4

x

–2

yn3 yn2 3y2 2 yn3 yn2 yn33y2 yn32 yn3n2 3yn32 2yn3 y2n1 3yn5 2yn3


–4

You Try It 4

2b2 15b 4 6b 15b2 13b 2 6b3 4b2 10b 8 6b3 15b2 12b 8 6b3 19b2 22b 8 3

x

You Try It 5

y

3 2 1 0 1 2 3 You Try It 6

You Try It 7

y

28 9 2 1 0 7 26

4 2 –4

–2

0

2

4

• 2(2b 2 5b 4) • 3b(2b 2 5b 4)

x

You Try It 5

–2

3x 42x 3 6x2 9x 8x 12 6x2 17x 12

–4

• FOIL

You Try It 6

3x2 14x 19 5x2 17x 11 8x2 11x 10

2 xn ynxn 4yn 2 x2n 8xnyn xnyn 4y2n 2 x2n 7xnyn 4y2n

Add the additive inverse of 6x2 3x 7 to 5x2 2 x 3.

You Try It 7

15x 2 x 3 16x2 3x 7 11x2 1x 4

You Try It 8

3x 73x 7 9x2 49

• FOIL

• The sum and difference of two terms

2

You Try It 8

Sx 4x3 3x2 2 2 x2 2 x 3 4x3 5x2 2 x 1 S1 41 51 21 1 41 51 2 1 4 5 2 1 12 3

ence of two terms

3x 4y2 • The square 9x2 24xy 16y2 of a binomial

You Try It 10

2xn 82 4x2n 32 xn 64

2

• The square of a binomial


Dx Px Rx


• The sum and differ-

You Try It 9

You Try It 9

Dx 5x2n 3xn 7 2 x2n 5xn 8 5x2n 3xn 7 2 x2n 5xn 8 7x2n 2 xn 15

Solution


2b2 7b 85b 2b25b 7b5b 85b • Use the Distributive 10b3 35b2 40b Property. You Try It 2

2xn 32 xn 3 4x2n 9

x2 2 x x x4x 5 x2

x2 2 x x 4x2 5x x2

x2 2 x 6x 3x2

x2 12 x2 6x3 6x3 11x2

To find the area, replace the variables 1 b and h in the equation A bh by 2 the given values and solve for A. 1 bh 2 1 A 2x 6x 4 2 A x 3x 4 A x2 4x 3x 12 A x2 x 12 A

• FOIL

The area is x2 x 12 square feet. You Try It 12 Strategy

To find the volume, subtract the volume of the small rectangular solid from the volume of the large rectangular solid. Large rectangular solid: Length L1 12x Width W1 7x 2 Height H1 5x 4

Chapter 6

Small rectangular solid: Length L2 12x Width W2 x Height H2 2x Solution

V Volume of large rectangular solid volume of small rectangular solid V L1 W1 H1 L2 W2 H2 V 12x7x 25x 4 12xx2x 84x2 24x5x 4 12x22x 420x3 336x2 120x2 96x 24x3 396x3 216x2 96x The volume is 396x3 216x2 96x cubic feet. You Try It 13 Strategy

To find the area, replace the variable r in the equation A r 2 by the given value and solve for A.

Solution

A r2 A 3.142x 32 3.144x2 12x 9 12.56x2 37.68x 28.26 The area is 12.56x2 37.68x 28.26 square centimeters.

3x3 8x2 6x 2 1 x2 3x 1 3x 1 3x 1 You Try It 4

3x2 12x 4 x 3x 2)3x 11x 16x2 16x 8 3x4 19x3 16x2 2

4

3

2x3 10x2 16x 2x3 16x2 14x 4x2 12x 8 4x2 12x 8 0 3x4 11x3 16x2 16x 8 3x2 2 x 4 x2 3x 2 You Try It 5

2

8 12

5 8

6

4

3

6x 8x 5 x 2 2

6x 4 You Try It 6

2

3 x2

5

12 10

8 4

16 24

5

2

12

8

5x3 12x2 8x 16 x 2

SECTION 6.4

5x2 2x 12

You Try It 1

4x3y 8x2y2 4xy3 4x3y 8x2y2 4xy3 2xy 2xy 2xy 2xy 2x2 4xy 2y2

You Try It 7

3

Check: 2xy2x2 4xy 2y2 4x3y 8x2y2 4xy3 You Try It 2

6

You Try It 8

2

16 15x 17x 20 16 5x 1 3x 4 3x 4 2

x2 3x 1 3x 1)3x3 8x2 6x 2 3x3 1x2 9x2 6x 9x2 3x 3x 2 3x 1 1

3 6

8 9

0 3

2 9

2

3

1

3

7

2

3 4

5 2

2

1

3

P2 3 You Try It 9

You Try It 3

2

2x4 3x3 8x2 2 x 3 7 2x3 3x2 x 3 x3

5x 11 3x 4)15x2 17x 20 15x2 20x 3x 20 3x 14

8 x2

3

2

5 6

0 33

7 99

2

11

33

92

P3 92


S20


S21


You Try It 2

The GCF is 7a2.

You Try It 1

14a2 21a4b 7a22 7a23a2b 7a22 3a2b The GCF is 9.

You Try It 2

27b2 18b 9 93b2 92b 91 93b2 2b 1

Factors

Sum

1, 1, 2, 2, 3, 3,

17 17 7 7 3 3

18 18 9 9 6 6

x2 7x 18 x 9x 2 You Try It 3

You Try It 3

The GCF is 2x.

2 2

The GCF is 3x y . 6x4y2 9x3y2 12x2y4 3x2y22x2 3x2y23x 3x2y24y2 3x2y22x2 3x 4y2 You Try It 4

2y5x 2 32 5x 2y5x 2 35x 2 5x 22y 3

• 5x 2 is the common factor.

2x3 14x2 12x 2xx2 7x 6 Factor the trinomial x2 7x 6. Find two negative factors of 6 whose sum is 7. Factors

Sum

1, 6 2, 3

7 5

2x3 14x2 12x 2xx 6x 1

You Try It 5

a2 3a 2ab 6b a2 3a 2ab 6b aa 3 2ba 3 • a 3 is the common factor. a 3a 2b

You Try It 4

The GCF is 3. 3x2 9xy 12y2 3x2 3xy 4y2 Factor the trinomial.

You Try It 6

2mn n 8mn 4 2mn2 n 8mn 4 n2mn 1 42mn 1 2mn 1n 4 2

• 2mn 1 is the common factor.

You Try It 7


Find the factors of 18 whose sum is 7.

3xy 9y 12 4x 3xy 9y 12 4x 3yx 3 43 x 3yx 3 4x 3 x 33y 4

Find the factors of 4 whose sum is 3.

Factors

Sum

1, 4 1, 4 2, 2

3 3 0

3x2 9xy 12y2 3x yx 4y • 12 4x (12 4x) • (3 x) (x 3) • x 3 is the common factor.


Factor the trinomial 2x2 x 3.

SECTION 7.2

Positive factors of 2: 1, 2

Factors of 3: 1, 3 1, 3

You Try It 1

Find the positive factors of 20 whose sum is 9.

Factors

Sum

Trial Factors

Middle Term

1, 20 2, 10 4, 5

21 12 9

x 12x 3 x 32x 1 x 12x 3 x 32x 1

3x 2x x x 6x 5x 3x 2x x x 6x 5x

x2 9x 20 x 4x 5

2x2 x 3 x 12x 3

Chapter 7

You Try It 2

The GCF is 3y. 45y3 12y2 12y 3y15y2 4y 4 Factor the trinomial 15y2 4y 4. Positive factors of 15: 1, 15 3, 5

Factors of 4:

Trial Factors y 115y 4 y 415y 1 y 115y 4 y 415y 1 y 215y 2 y 215y 2 3y 15y 4 3y 45y 1 3y 15y 4 3y 45y 1 3y 25y 2 3y 25y 2

1, 4 1, 4 2, 2

Middle Term 4y 15y 11y y 60y 59y 4y 15y 11y y 60y 59y 2y 30y 28y 2y 30y 28y 12y 5y 7y 3y 20y 17y 12y 5y 7y 3y 20y 17y 6y 10y 4y 6y 10y 4y

45y3 12y2 12y 3y3y 25y 2 You Try It 3 Factors of 14 [2(7)] 1, 14 1, 14 2, 7 2, 7

Sum 13 13 5 5

3x2 8x 16 3x2 4x 12x 16 3x2 4x 12x 16 x3x 4 43x 4 3x 4x 4 15x3 40x2 80x 5x3x2 8x 16 5x3x 4x 4


x2 36y4 x2 6y22 x 6y2x 6y2

• Difference of two squares

You Try It 2

9x2 12x 4 3x 22

• Perfect-square trinomial

You Try It 3

• Difference of two squares a b2 a b2 a b a b a b a b

a b a ba b a b 2a2b 4ab You Try It 4

• Difference of two cubes a3b3 27 ab3 33 ab 3a2b2 3ab 9 You Try It 5

8x3 y3z3 2x3 yz3 • Sum of two cubes 2x yz4x2 2xyz y2z2 You Try It 6

2a2 13a 7 2a 1a 7

• Sum of two cubes x y3 x y3 x y x y

x y2 x yx y x y2

2 x x2 2 xy y2 x2 y2 x2 2 xy y2

2 xx2 2 xy y2 x2 y2 x2 2 xy y2 2 xx2 3y2

You Try It 4

You Try It 7

2a2 13a 7 2a2 a 14a 7 2a2 a 14a 7 a2a 1 72a 1 2a 1a 7

The GCF is 5x.

3x4 4x2 4 3u2 4u 4 u 23u 2 x2 23x2 2

15x3 40x2 80x 5x3x2 8x 16

Factors of 48 [3(16)] 1, 1, 2, 2, 3, 3, 4, 4,

48 48 24 24 16 16 12 12

Sum 47 47 22 22 13 13 8 8

Let u x2.

You Try It 8

18x3 6x2 60x 6x3x2 x 10 6x3x 5x 2 You Try It 9

• GCF

4x 4y x3 x2y • Factor by grouping. 4x 4y x3 x2y 4x y x2x y x y4 x2 x y2 x2 x


S22


The sum of the squares of the two consecutive positive integers is 61.

You Try It 10

x4n x2ny2n x2n2n x2ny2n x2nx2n y2n x2n xn2 yn2

x2nxn ynxn yn

• GCF • Difference of

Solution

n2 n 12 61 n n2 2n 1 61 2n2 2n 1 61 2n2 2n 60 0 2n2 n 30 0 2n 5n 6 0 2

two squares

You Try It 11

ax5 ax2y6 ax2x3 y6 • GCF ax2x y2x2 xy2 y4 • Difference of two cubes

n50 n5

SECTION 7.5

n 6 0 • Principle of n 6 Zero Products

Because 6 is not a positive integer, it is not a solution.

You Try It 1

2xx 7 0 2x 0 x0

x70 x 7

n5 n1516

• Principle of Zero Products

The two integers are 5 and 6.

The solutions are 0 and 7. You Try It 5

You Try It 2

4x2 9 0 2x 32x 3 0

• Difference of two squares

2x 3 0 2x 3 3 x 2


2x 3 0 2x 3 3 x 2

The solutions are

Strategy

The area of the rectangle is 96 in2. Use the equation A L W . Solution

3 3 and . 2 2

x 2x 7 52 x2 5x 14 52 x2 5x 66 0 x 6x 11 0 x60 x 6

Width x

Length 2x 4

You Try It 3


S23

ALW 96 2x 4x 96 2x2 4x 0 2x2 4x 96 0 2x2 2x 48 0 2x 8x 6 x80 x 8

x 11 0 x 11

x60 x6


Because the width cannot be a negative number, 8 is not a solution.



x6 2x 4 26 4 12 4 16

You Try It 4

The length is 16 in. The width is 6 in. First consecutive positive integer: n Second consecutive positive integer: n1

Strategy


You Try It 2

You Try It 1

x 4x 6 x 2 2x 24 16 x 2 4 x4 x

1

1

1

6x y 23xy x 2 3 2 3 12x y 223xy 2y2 5

5

1

1

1

3

x6 x4

• (4 x) 1(x 4)

S24

Chapter 8

The LCM is xx 32x 5.

You Try It 3 1

3x x 3x 2 3x 2x 11x 15 x 32x 5 x xx 32x 5

x 2x 6 x6 x2 4x 12 x2 3x 2 x 1x 2 x1

2

x2 x2 2x 5 2x2 9x 10 2 x 3x xx 3 2x 5 xx 32x 5

1

You Try It 4

3x4x 1 42x 3 12x2 3x 8x 12 10x 15 9x 18 52x 3 9x 2 1

• Factor.

1

3x4x 1 2 22x 3 52x 3 3 3x 2 1

1

You Try It 3

2x x2 x2 x xx 2; 3x2 5x 2 x 23x 1 The LCM is xx 23x 1. 2x 7 3x 1 6x2 19x 7 2x 7 2 2x x xx 2 3x 1 xx 23x 1

4x4x 1 15x 2

3x 2 3x 2 x 3x 2 2x 3x 5x 2 x 23x 1 x xx 23x 1 2

You Try It 5

1

1

You Try It 4

The LCM is ab. • Factor.

1

x 3x 5 x 3x 6 x5 3 x3 x x 1x 6 x1 1

1

1

2 1 4 2 a 1 b a ab b a a 2a b ab ab

b 4 b ab 4 2a b 4 ab ab

You Try It 6

You Try It 5

a2 a 2 2 4bc 2b c 6bc 3b2 2 6bc 3b2 a 4bc 2 2b2c a

The LCM is aa 5a 5. a3 a9 2 a2 5a a 25 a3 a5 a9 a aa 5 a 5 a 5a 5 a a 3a 5 aa 9 aa 5a 5 a2 2a 15 a2 9a aa 5a 5 a2 2a 15 a2 9a aa 5a 5 2a2 7a 15 2a 3a 5 aa 5a 5 aa 5a 5

1

• Multiply by the reciprocal.

1

a2 3b2c b 3a 2bc2c b a 2c 1

1

You Try It 7

3x2 26x 16 2x2 9x 5 2 2 3x 7x 6 x 2x 15 2 3x 26x 16 x2 2x 15 2 3x2 7x 6 2x 9x 5 1

1

• Multiply by the reciprocal.

1

x8 3x 2x 8 x 5x 3 3x 2x 3 2x 1x 5 2x 1 1

1

1

1

2a 3 2a 3a 5 aa 5a 5 aa 5 1

SECTION 8.2

You Try It 6

You Try It 1

2x x1 2 2 x4 x1 x 3x 4 2x x1 x1 x4 2 x4 x1 x1 x4 x 4x 1 2xx 1 x 1x 4 2 x 4x 1 2x2 2x x2 5x 4 2 x 4x 1 x2 7x 2 x 4x 1

The LCM is 2x 5x 4. 2x x4 2x2 8x 2x 2x 5 2x 5 x 4 2x 5x 4 3 3 2x 5 6x 15 x4 x 4 2x 5 2x 5x 4 You Try It 2

2x2 11x 15 x 32x 5; x2 3x xx 3

The LCM is x 4x 1.


x2 2x 15 x2 3x 18 2 9 x2 x 7x 6 x 3x 5 x 3x 6 3 x3 x x 1x 6

S25


SECTION 8.3

You Try It 2

You Try It 1

The LCM is x 3. 14 14 2x 5 2x 5 x3 x3 x3 49 49 x3 4x 16 4x 16 x3 x3 14 2x 5x 3 x 3 x3 49 4x 16x 3 x 3 x3 2x2 x 15 14 2x2 x 1 2 4x 4x 48 49 4x2 4x 1 1

2x 1x 1 2x 1x 1 x1 2x 12x 1 2x 12x 1 2x 1

10 5x 3 x2 x2 5x x2 10 x2 3 1 x2 1 x2

1

• The LCM is x 2. • Clear denominators.

1

x2 5x x2 x2 10 3 1 x2 1 1 x2 1

1

5x 5x 5x 2x x

x 23 10 3x 6 10 3x 4 4 2

• Solve for x.

2 does not check as a solution. The equation has no solution.

1

SECTION 8.5

You Try It 2

The LCM of the denominators is x. 1

1

x 2 2 1 1 x 2 2 x x 1x x 2 2 2x 1 1 2x x x The LCM of the denominators is 2x 1.

2 6 x3 5x 5 x 35x 5 2 x 35x 5 6 1 x3 1 5x 5 1

1

x 35x 5 x 35x 5 2 6 1 x3 1 5x 5 1

x 2x 1 x 2 2 2x 1 2x 1 2x 1 4x 2 x 2x 1 2x 1 4x 2 x 3x 2 2x 1 2x 1

The solution is 2.

SECTION 8.4

You Try It 2

x 3 x6 x

• The LCM is x(x 6).

xx 6 x xx 6 3 1 x6 1 x

• Multiply by the LCM.

1

x30 x 3

x60 x6

To find the total area that 256 ceramic tiles will cover, write and solve a proportion using x to represent the number of square feet that 256 tiles will cover.

• Factor. • Principle of Zero Products

An area of 144 ft 2 can be tiled using 256 ceramic tiles.

Solution

x x 63 x2 3x 18 x2 3x 18 0 x 3x 6 0

• Solve for x.

9 x 16 256 9 x 256 256 16 256 144 x

1

2

1

5x 52 x 36 10x 10 6x 18 4x 10 18 4x 8 x2

Strategy

You Try It 1 Copyright © Houghton Mifflin Company. All rights reserved.

You Try It 1

• Simplify.

Both 3 and 6 check as solutions. The solutions are 3 and 6.

• Write a proportion. • Clear denominators.

Chapter 8


Solution

CD DO • Write a proportion. AB AO 4 3 • Substitute. 10 AO 4 3 10 AO 10 AO 10 AO 4AO 30 AO 7.5 1 bh 2 1 107.5 2 37.5

A

• Area of a triangle

S rS S rS 1 rS 1 rS 1r

rS rS C C C C 1r C S 1r

Strategy

You Try It 3

AL 2 AL 2s2 2 2s A L 2s A A A L 2s A L

• Time for one printer to complete the job: t Rate

Time

1 t 1 t

1st printer

• Substitute.

2 5

Part 2 t 5 t

• The sum of the parts of the task completed must equal 1.

5x 2y 10 5x 5x 2y 5x 10 2y 5x 10 2y 5x 10 2 2 5 y x5 2 s

• Factor. • Divide by 1 r.

You Try It 1

Solution

SECTION 8.6

You Try It 2

• Subtract rS.

SECTION 8.7

2nd printer

The area of triangle AOB is 37.5 cm2.

You Try It 1

S rS C

You Try It 4

To find the area of triangle AOB: • Solve a proportion to find the length of AO (the height of triangle AOB). • Use the formula for the area of a triangle. AB is the base and AO is the height.

• Subtract 5x. • Divide by 2.

5 2 1 t t 2 5 t1 t t t 25t 7t

Working alone, one printer takes 7 h to print the payroll. You Try It 2 Strategy

• Rate sailing across the lake: r Rate sailing back: 3r

• Multiply by 2.

Distance

Rate

Across

6

r

Back

6

3r

• Subtract A.

S a n 1d S a nd d S a a a nd d S a nd d S a d nd d d S a d nd Sad nd d d Sad n d

Time 6 r 6 3r

• The total time for the trip was 2 h. Solution • Subtract a. • Add d. • Divide by d.

6 6 2 r 3r 6 6 3r2 • Multiply by 3r r 3r the LCM, 3r. 6 6 6r 3r 3r r 3r 18 6 6r • Solve for r. 24 6r 4r

The rate across the lake was 4 kmh.


S26


SECTION 8.8


You Try It 1

• Write the basic direct variation equation, replace the variables by the given values, and solve for k. • Write the direct variation equation, replacing k by its value. Substitute 5 for t and solve for s. s kt2 64 k22 64 k 4 16 k

Solution

• Direct variation equation • Replace s by 64 and t by 2.

s 16t2 1652 400 • k 16, t 5 The object will fall 400 ft in 5 s. You Try It 2

To find the resistance:

Strategy

• Write the basic inverse variation equation, replace the variables by the given values, and solve for k. • Write the inverse variation equation, replacing k by its value. Substitute 0.02 for d and solve for R.

To find the strength of the beam: • Write the basic combined variation equation, replace the variables by the given values, and solve for k. • Write the combined variation equation, replacing k by its value and substituting 4 for W, 8 for d, and 16 for L. Solve for s.

To find the distance:

Strategy

Solution

kWd2 L k2122 1200 12 1200 24k 50 k 50Wd2 s L 50482 16 800 s

• Combined variation equation • Replace s by 1200, W by 2, d by 12, and L by 12.

• Replace k by 50 in the combined variation equation.

• Replace W by 4, d by 8, and L by 16.

The strength of the beam is 800 lb.

Solution

R

k d2

• Inverse variation equation

k 0.012 k 0.5 0.0001 0.00005 k 0.5


R

S27

• Replace R by 0.5 and d by 0.01.

0.00005 0.00005 0.125 d2 0.022

• k 0.00005, d 0.02

The resistance is 0.125 ohm.

Solutions to Chapter 9 “You Try It” denominator of the exponent is a positive even number.


1634 2434 23 1 1 3 2 8

You Try It 2

8134 The base of the exponential expression is negative, and the

Therefore, 8134 is not a real number.

You Try It 3

x34y12z2343 x1y23z89 z89 23 xy

S28

Chapter 9

You Try It 4

16a2b43 9a4b23

12

24a6b2 32

22a3b1 31

You Try It 5

12

• Use the Rule for Dividing Exponential Expressions. • Use the Rule for Simplifying Powers of Products.

3a3 3a3 2 2b 4b You Try It 5

4 2x334 2x33 4 8x9

3 3 2 2x 3 2x 5 3 3 3 2 2 4x 10 2x 3 2x 15 • The FOIL method 3 3 2 4x2 13 2x 15

You Try It 6

a 3y a 3y a 2 3y 2 a 9y

5a56 5a516 6 5 a5

You Try It 7

3ab 3ab13

are conjugates.

You Try It 7

y 3y

You Try It 6

• The expressions

y 3y

3y 3y

y3y

• Rationalize the

9y

denominator.

2

y3y 3y 3y 3 You Try It 8

3

3x 3x2 3

4

You Try It 8

x4 y4 x4 y414

You Try It 9

8x12y3 2x4y

3x

3 3 3x9x 9x 3 3 9x 27x3

• Rationalize the

3x2 3 3 3x9x 9x 3x 3

denominator.

3

You Try It 9

3 6

4

81x12y8 3x3y2

2 6

SECTION 9.2 5 5 x7 x5 x2

You Try It 1

5 5 5 2 x x 5 2 x x

• x 5 is a perfect

2 6

• Rationalize the

2 6 2 6 6 36 26 6 2

denominator.

22 6 2 12 56 6 56 6 46 2 12 56 2

SECTION 9.3 3

64x y 64x y x 3 3 64x6y18 x2 3 4x2y6 x2 8 18

fifth power.

You Try It 2 3

3 6

6 18

2

• 64x 6y 18 is a perfect

You Try It 1

3 3 3xy 81x5y 192 x8y4 3 3 3 3xy27x 3x2y 64x6y3 3x2y 3 3 3 3 3xy27x3 3x2y 64x6y3 3x2y 3 3 3xy 3x3x2y 4x2y3x2y 3 2 3 2 3 2 9x y 3x y 4x y 3x2y 5x2y 3x2y

• Add x 5 x x 5 1 to each side. 1 5 x x 2 2 • Square each side. x 1 x 5 x 1 2x 5 x 5 6 2x 5 3 x 5 • Square each side. 32 x 5 2 9x5 4x

You Try It 4

4 does not check as a solution. The equation has no solution.

third power.

You Try It 3

5b 3b 10 15b2 50b b2 15 25 2b b2 15 25 2b b15 52b

• The Distributive Property • Simplify each radical expression.

You Try It 2

4

x 8 3 x 8 4 34 x 8 81 x 89 4

• Raise each side to the fourth power.


You Try It 10


S29

SECTION 9.4

Check: 4 x 8 3 4 3 89 8 4 3 81 33

You Try It 1

45 i45 i9 5 3i5

You Try It 2

98 60 98 i60 49 2 i4 15 72 2i15

You Try It 3

4 2i 6 8i 10 10i

You Try It 4

16 45 3 20 16 i45 3 i20 16 i9 5 3 i4 5 16 3i5 3 2i5 13 5i5

You Try It 5

3 2i 3 2i 0 0i 0

You Try It 6

3i10i 30i2 301 30

The solution is 89. You Try It 3 Strategy

To find the diagonal, use the Pythagorean Theorem. One leg is the length of the rectangle. The second leg is the width of the rectangle. The hypotenuse is the diagonal of the rectangle.

Solution

c2 a2 b2 c2 62 32 c2 36 9 c2 45 c212 4512 c 45 c 6.7

• Pythagorean Theorem • Replace a by 6 and b by 3. • Solve for c. 1 2

• Raise each side to the power. • a a 1/2

The diagonal is approximately 6.7 cm.

You Try It 7

8 5 i8 i5 i240 140 4 10 210



Solution

To find the height, replace d in the equation with the given value and solve for h. d 1.5h 5.5 1.5h 5.52 1.5h 2 30.25 1.5h 20.17 h

• Replace d by 5.5. • Square each side.

Solution

• The Distributive Property

You Try It 9

The periscope must be approximately 20.17 ft above the water.

327 6 i327 i6 i81 i218 i81 19 2 9i 32 32 9i

To find the distance, replace the variables v and a in the equation by the given values and solve for s.

You Try It 10


You Try It 8

6i3 4i 18i 24i2 18i 241 24 18i

v 2as 88 2 22s 88 44s 882 44s 2 7744 44s 176 s

• Replace v by 88 and a by 22.

4 3i2 i 8 4i 6i 3i2 8 10i 3i2 8 10i 31 5 10i

• FOIL

• Square each side.

The distance required is 176 ft.

You Try It 11

3 6i3 6i 32 62 • Conjugates 9 36 45

S30

Chapter 10

You Try It 12

2 3i 2 3i i 4i 4i i 2i 3i2 4i2 2i 31 41 3 2i 3 1 i 4 4 2

You Try It 13

2 5i 3 2i 6 4i 15i 10i2 2 5i 3 2i 3 2i 3 2i 32 22 4 19i 6 19i 101 94 13 19 4 i 13 13


You Try It 4

You Try It 1

2x 12 24 0 2x 12 24 x 12 12

2x2 7x 3 2 2x 7x 3 0 2x 1x 3 0 2x 1 0 2x 1

• Write in standard form. • Solve by factoring.

x30 x3

• Take the square root of each side of the equation.

x 1 23 • Solve for x. x 1 23

The solutions are 1 23 and 1 23. 1 and 3. 2

SECTION 10.2

x2 3ax 4a2 0 x ax 4a 0

You Try It 2

xa0 x a

x 4a 0 x 4a

The solutions are a and 4a.

You Try It 3

x r1x r2 0

1 x 3 x 2

1 x 3 x 2 x2

0

x

20

2x2 5x 3 0

1 2

x

0

5 3 x 0 2 2

5 3 x 2 2

1 • r1 3, r2 2

You Try It 1

4x2 4x 1 0 4x2 4x 1 1 1 4x2 4x 1 4 4 1 2 x x 4 1 1 1 x2 x 4 4 4 1 2 2 x 2 4

x

2

2 4

• Write in the form ax 2 bx c. 1 a

• Multiply both sides by .

• Complete the square. • Factor. • Take square roots.

1 2 2 2

1 2 2 2 1 2 x 2 2

The solutions are

x

1 2 2 2 1 2 x 2 2

1 2 1 2 and . 2 2

• Solve for x .


The solutions are

x 1 12 x 1 23 x 1 23 x 1 23

1 x 2

2 x2

x 12 12

• Solve for (x 1)2.


SECTION 10.4

You Try It 2

x2 4x 8 0 x2 4x 8 2 x 4x 4 8 4 x 22 4 x 22 4 x 2 2i x 2 2i x 2 2i

You Try It 1

• Complete the square. • Factor. • Take square roots.

x 2 2i x 2 2i

u20 u2

• Solve for x .

The solutions are 1 3i and 1 3i. You Try It 2

4x2 4x 1 • Write in standard form. 4x 4x 1 0 a 4, b 4, c 1 b b2 4ac x 2a 4 42 441 24 4 0 4 16 16 8 8 4 1 8 2


2

The solution is

You Try It 3

1 . 2

3x2 x 1 0 a 3, b 1, c 1 b2 4ac 12 431 1 12 13 13 0 Because the discriminant is greater than zero, the equation has two real number solutions.

u3 0 u3

Replace u by x12. x12 2 x 2 x 2 22 x4

SECTION 10.3 x2 2x 10 0 a 1, b 2, c 10 b b2 4ac x 2a 2 22 4110 21 2 4 40 2 36 2 2 2 6i 1 3i 2

x 5x12 6 0 x 5x12 6 0 u2 5u 6 0 u 2u 3 0 12 2

The solutions are 2 2i and 2 2i.

You Try It 1

S31

x12 3 x 3 x 2 32 x9

4 and 9 check as solutions. The solutions are 4 and 9. You Try It 2

2x 1 x 7 2x 1 7 x 2x 1 2 7 x2 2x 1 49 14x x2 0 x2 16x 48 0 x 4x 12 x40 x4

• Solve for the radical.

• Square each side. • Write in standard form.

• Solve by factoring.

x 12 0 x 12

4 checks as a solution. 12 does not check as a solution. The solution is 4. You Try It 3

2x 1 x 2 Solve for one of the radical expressions. 2x 1 2 x 2x 1 2 2 x 2 2x 1 4 4x x x 5 4x x 52 4x 2 x2 10x 25 16x x2 26x 25 0 x 1x 25 0 x10 x1

x 25 0 x 25

1 checks as a solution. 25 does not check as a solution. The solution is 1.



S32

Chapter 11

SECTION 10.6

You Try It 4

25 3y 8 3y 2 25 3y 28 3y 2 3y 3y 2


• Clear denominators.

25 3y 28 3y 23y 3y 2 3y 2 9y2 6y 25 24y 16

• This is a geometry problem. • Width of the rectangle: W Length of the rectangle: W 3 • Use the equation A L W . ALW

Solution

54 W 3W

• Write in standard form.

54 W 2 3W

9y2 18y 9 0 9 y2 2y 1 0

0 W 2 3W 54 0 W 9W 6

• Solve by factoring. 9 y 1 y 1 0 y10 y 1

W90 W 9

y10 y 1

The solution 9 is not possible.

The solution is 1.

W3639 The length is 9 m.


W60 W6

2x2 x 10 0 2x 5x 2 0 2x − 5 − − − − − − − − − − − − − + + + x + 2 −−− ++++++++++ +++ −3 −2 −1

0

1

2

3

−5 − 4 −3 −2 −1 0 1 2 3 4 5

x 2 x

5 2

Solutions to Chapter 11 “You Try It” You Try It 2

You Try It 1

x-coordinate of vertex: 2 b 1 2a 21

x-coordinate of vertex: b 4 1 2a 24 2

y-coordinate of vertex: y 4

y-coordinate of vertex: y 4x2 4x 1 1 2 1 4 4 2 2 121 0

2

1

–4 –2 0 –2 –4

2

4

x

fx x2 2x 1 f1 12 21 1 1 2 1 0

y 4 2 –4 –2 0 –2

2

4

x

–4

Vertex: 1, 0 The domain is xx real numbers . The range is y y 0 .

1 Vertex: , 0 2

You Try It 3

1 Axis of symmetry: x 2

y x2 3x 4 0 x2 3x 4


SECTION 11.1


b b2 4ac 2a 3 32 414 21 3 7 2 3 i7 2

S33

• To find the maximum height, evaluate the function at the t-coordinate of the vertex.

x

• a 1, b 3, c 4 Solution

t

• The t-coordinate

64 b 2 2a 216

of the vertex

The ball reaches its maximum height in 2 s.

The equation has no real number solutions. There are no x-intercepts.

st 16t2 64t s2 1622 642 64 128 64

You Try It 4

The maximum height is 64 ft.

gx x x 6 0 x2 x 6

•t2

2

You Try It 8

b b 4ac 2a 1 12 416 21 1 1 24 2 1 i23 1 23 2 2

x

2

The zeros of the function are 1 23 i. 2 2

Strategy

• a 1, b 1, c 6

You Try It 5

Solution


b2 4ac 12 416 1 24 25

• P 100 • Solve for y.

Express the area of the rectangle in terms of x. A xy A x100 2x • y 100 2x A 2x2 100x • To find the width, find the x-coordinate of the vertex of fx 2x2 100x. • To find the length, replace x in y 100 2x by the x-coordinate of the vertex.

1 23 i and 2 2

y x2 x 6 a 1, b 1, c 6

P 2x y 100 2x y 100 2x y

x

b 100 25 2a 22

The width is 25 ft.

Because the discriminant is greater than zero, the parabola has two x-intercepts.

100 2x 100 225 100 50 50

You Try It 6

The length is 50 ft.

fx 3x2 4x 1 4 2 b x 2a 23 3 fx 3x2 4x 1 2 2 2 2 f 3 4 1 3 3 3 4 8 1 1 3 3 3

• The x-coordinate of the vertex

• x

2 3

Because a is negative, the function has a maximum value. The maximum value of the function is


Any vertical line intersects the graph at most once. The graph is the graph of a function. y

You Try It 2 4

1 . 3

2 –4

0

2

4

x

–2


–2

• To find the time it takes the ball to reach its maximum height, find the t-coordinate of the vertex.

–4

Domain: x x real numbers Range: y y real numbers

S34

Chapter 11

y

You Try It 3

You Try It 4

gx x2 g1 12 1

4 2 –4

–2

0

2

4

x

–2 –4

Domain: x x real numbers Range: y y 0

4 2 –4

–2

0

fx 1 2x f g1 f1 1 21 1 f g1 1

• Evaluate f at g (1) 1.

You Try It 5

Ms s3 1

• Evaluate M at L(s).

M Ls s 1 1 s3 3s2 3s 1 1 s3 3s2 3s 2 M Ls s3 3s2 3s 2 3

y

You Try It 4

• Evaluate g at 1.

2

4

x

–2 –4



f g2 f2 g2 22 22 52 2

4 4 10 2 12

You Try It 2

f g2 12 You Try It 2

f g3 f3 g3 4 32 33 4

4 9 9 4 55 25

f f4 4 g g4 42 4 42 2 4 1 16 4 16 8 1 12 25

f 12 4 g 25

1 x4 2 1 y x4 2 1 x y4 2 1 x4 y 2 2x 8 y f1x 2x 8 fx

The inverse of the function is given by f1x 2x 8.

f g3 25 You Try It 3

Because any horizontal line intersects the graph at most once, the graph is the graph of a 1–1 function.

You Try It 3

1 x3 6 2 x 6 6 x 12

f gx 2

No, gx is not the inverse of fx. Copyright © Houghton Mifflin Company. All rights reserved.

Domain: x x 1 Range: y y 0

S35



You Try It 7

x 4

You Try It 1

fx f3 f2

2 3

x

2 3

3

2 3

2

2

8 27

2

3 2

0 2 4

• x3

9 4

f2 2

2

• x0

1 1 3 2 8

You Try It 1

• x 2

fx e2x1 f2 e221 e3 20.0855 f2 e221 e5 0.0067 x

y

4 2 0

4 2 1 1 2 1 4

2 4

You Try It 5

x 2


1 0 1 2

You Try It 6

x

y

2 1 0

6 4 3 5 2 9 4

1 2

–4

–2

0

2

4

x

–2 –4

You Try It 2

4 2 –2

0

2

4

• Write an equation. • Write the equivalent • 64 43 • The bases are the same. The exponents are equal.

log464 3

y

–4

log464 x 64 4x

exponential form.

• x2 • x 2

x

log2 x 4 24 x 1 x 24 1 x 16

• Write the equivalent exponential form.

–2

The solution is

–4

You Try It 3 y

y 5 4 3 2 2 3 5

2

43 4x 3x

You Try It 3

You Try It 4

9 4 5 2 3 4 6

SECTION 12.2

fx 22x1 f0 2201 21 2 3

4

• x 2

You Try It 2

22 1

y

y

1 . 16

ln x 3 e3 x • Write the equivalent exponential form. 20.0855 x

4

You Try It 4

2 –4

–2

0

2

4

x

3 log8 xy2 log8xy213

–2

1 log8xy2 • Power Property 3

1 log8 x log8 y2 3 1 log8 x 2 log8 y 3 1 2 log8 x log8 y 3 3

–4

y

• Product Property • Power Property • Distributive Property

4

You Try It 5

2 –4

–2

0 –2 –4

2

4

x

1 log4 x 2 log4 y log4 z 3 1 log4 x log4 y2 log4 z 3 1 x 1 xz log4 2 log4 z log4 2 3 y 3 y 13 xz xz log4 2 log4 3 2 y y

Chapter 12

You Try It 6

Because log b1 0, log91 0.

You Try It 7

log30.834

You Try It 8

SECTION 12.4

ln 0.834 0.16523 ln 3

log 6.45 0.95795 log 7

log76.45


You Try It 2

exponential equation. y 4

2

2

1

–4

1 18 1 6 1 2 3 2 9 2

–2

0

2

4

x

–2

0 1 2

–4

• Divide both sides by log 1.06.

4 25 log 43x log 25 3x log 4 log 25 log 25 3x log 4 3x 2.3219 x 0.7740

• Take the log of each side. • Power Property • Divide both sides by log 4. • Divide both sides by 3.

The solution is approximately 0.7740. You Try It 3

fx log32x y log32x 3 y 2x 3y x 2 x

• Take the log of each side. • Power Property

• f (x) y • Write the equivalent You Try3x It 2

y

5 4 3 2 2 3 5

1.06n 1.5 log1.06n log 1.5 n log 1.06 log 1.5 log 1.5 n log 1.06 n 6.9585

The solution is approximately 6.9585.

fx log2x 1 y log2x 1 2y x 1 2y 1 x x

You Try It 1

log4x2 3x 1

• f (x) y • Write the equivalent exponential equation.

y

x10 x 1

y 4

2 1

–2

0

2

4

x

–4

You Try It 4

xx 3 4 x2 3x 4 2 x 3x 4 0 x 4x 1 0

1 2

x40 x 4

You Try It 3

fx log3x 1 y log3x 1 y log3x 1 3y x 1 y 3 1x

• f (x) y • Multiply both sides by 1. • Write the equivalent exponential equation.

x

y

8 2 0 2 3 8 9

2 1 0 1

y 4 2 –4

–2

0 –2 –4

2

x40 x4

log3 x log3x 3 log34 log3 xx 3 log34

–2

0

• Write in exponential form.


2 –4

41 x2 3x 4 x2 3x 0 x2 3x 4 0 x 1x 4

2

4

x

• Product Property • 1– 1 Property of Logarithms


x10 x1

4 does not check as a solution. The solution is 1. You Try It 5

log3 x log3x 6 3 log3 xx 6 3 xx 6 33 x2 6x 27 2 x 6x 27 0 x 9x 3 0 x90 x 9

• Product Property • Write in exponential form.


x30 x3

9 does not check as a solution. The solution is 3.


S36


I . Replace M I0

SECTION 12.5

earthquake, M log

You Try It 1

by 4.6 and solve for I.

Strategy

To find the hydrogen ion concentration, replace pH by 2.9 in the equation pH logH and solve for H.

Solution

pH logH 2.9 logH 2.9 logH 102.9 H 0.00126 H

• Multiply by 1. • Write the equivalent exponential equation.

S37

Solution M log

I I0

4.6 log

I I0

104.6

I I0

• Replace M by 4.6. • Write in exponential form.

104.6I0 I

The hydrogen ion concentration is approximately 0.00126.

39,811I0 I

You Try It 2

The earthquake had an intensity that was approximately 39,811 times the intensity of a zerolevel earthquake.

Strategy

To find the intensity, use the equation for the Richter scale magnitude of an

Solutions to Chapter R “You Try It” You Try It 5

SECTION R.1

a. 3(5y 2) 3(5y) (3)(2) 15y 6

You Try It 1

(4)(6 8)2 (12 4)

4(2)2 (3) 4(4) (3) 16 (3) 16 3 13


You Try It 2

3xy2 3x2y 3(2)(5)2 3(2)2(5) You Try It 3

3(2)(25) 3(4)(5) 6(25) 3(4)(5) 150 3(4)(5) 150 12(5) 150 60 150 (60) 210

5(3a) [5(3)]a 15a

You Try It 4

2z2 5z 3z2 6z 2z2 3z2 5z 6z (2z2 3z2) (5z 6z)

1z2 z

z2 z

b. 2(4x 2y 6z) 2(4x) (2)(2y) (2)(6z) 8x 4y 12z You Try It 6

a. 7(3x 4y) 3(3x y) 21x 28y 9x 3y 30x 31y b. 2y 3[5 3(3 2y)]

2y 3[5 9 6y] 2y 3[4 6y] 2y 12 18y 20y 12

SECTION R.2 You Try It 1

a.

4x 4x 7x 3x 3x 3

3 3 3 3 3x 3x 3 x

7x 9 7x 7x 9 9 93 6 6 3 2

The solution is 2.

S38

Chapter R

b.

4 (5x 8) 4x 3 4 5x 8 4x 3 5x 12 4x 3 5x 4x 12 4x 4x 3 9x 12 3 9x 12 12 3 12 9x 9 9x 9 9 9 x1

You Try It 3

3x 2y 4 2y 3x 4 3 y x2 2 y 4

x

y

2 0 2

5 2 1

2 –4

–2

0

2

4

x

–2 –4

The solution is 1. You Try It 4

3x 1 5x 7 3x 5x 1 5x 5x 7 2x 1 7 2x 1 1 7 1 2x 6 2x 6 2 2 x 3

m

You Try It 5

3 2(3x 1) 7 2x 3 6x 2 7 2x 1 6x 7 2x 1 6x 2x 7 2x 2x 1 4x 7 1 1 4x 7 1 4x 6 4x 6 4 4 3 x 2

xx

3 2

Place a dot at the y-intercept. Starting at the y-intercept, move to the right 3 units (the change in x) and down 2 units (the change in y). Place a dot at that location. Draw a line through the two points. y 4 2

–4

(0, 3)

2

(4, 0) –4

–2

0

(–4, –3) – 2

0

2

4

x

–4

You Try It 6 4

–2

–2

y (–2, 4)

2 y x2 3 2 2 m 3 3 y-intercept (0, 2)

SECTION R.3 You Try It 1

6 3 3 y2 y1 2 x2 x1 1 (2) 3

The slope is 2.

{x x 3} You Try It 3

Let (x1, y1) (2, 3) and (x2, y2) (1, 3).

2

4

f(x) 4 2x f(3) 4 2(3) f(3) 4 6 f(3) 10 An ordered pair of the function is (3, 10).

x

(5, –1)

–4


y

3 x4 5 y 4

x

y

5 0 5

1 4 7

2 –4

–2

0 –2 –4

2

4

x

(x1, y1) (2, 2), m y y1 m(x x1) 1 y 2 [x (2)] 2 1 y 2 (x 2) 2 1 y2 x1 2 1 y x1 2

1 2


You Try It 2


You Try It 7

SECTION R.4

(x3 7 2x) (x 2)

You Try It 1 a. (2ab)(2a3b2)3 (2ab)(23a9b6)

b.

2x2y4 4x2y5

3

22)a8b7 b7 2 8 2a b7 8 4a

x2y4 2x2y5

x2 2x 2 x 2) x 0x2 2x 7 x3 2x2 3

2x2 2x 2x2 4x 2x 7 2x 4

3

3

x2(3)y(4)(3) 1(3) (2)(3) (5)(3) 2 x y x6y12 3 6 15 2 xy 23x66y1215 23x12y3 8 12 3 x y

Check: (x 2)(x2 2x 2) 3 x3 2x 4 3 x3 2x 7 (x3 7 2x) (x 2) 3 x2 2x 2 x2

The GCF is 3x2y2.

You Try It 8

6x4y2 9x3y2 12x2y4 3x2y2(2x2 3x 4y2)

You Try It 2

(4x3 2x2 8) (4x3 6x2 7x 5) (4x3 4x3) (2x2 6x2) (7x) (8 5) 8x2 7x 3

Two factors of 15 whose sum is 8 are 3 and 5.

You Try It 9

x2 8x 15 (x 3)(x 5)

You Try It 3

(4w 8w 8) (3w 4w 2w 1) (4w3 8w 8) (3w3 4w2 2w 1) (4w3 3w3) 4w2 (8w 2w) (8 1) 7w3 4w2 6w 7 3

3

2

You Try It 4


3mn2(2m2 3mn 1) 3mn2(2m2) 3mn2(3mn) 3mn2(1) 6m3n2 9m2n3 3mn2

You Try It 5

S39

3c2 4c 5 2c 3

9c 12c 15 6c3 8c2 10c 2

6c3 17c2 22c 15

You Try It 10

3x2 x 2 Positive Factors of 3

Factors of 2

1, 3

1, 2 1, 2

Trial Factors

Middle Term

x 13x 2 x 23x 1 x 13x 2 x 23x 1

2x 3x x x 6x 5x 2x 3x x x 6x 5x

3x2 x 2 (x 1)(3x 2) You Try It 6

(4y 7)(3y 5) (4y)(3y) (4y)(5) (7)(3y) (7)(5) 12y2 20y 21y 35 12y2 41y 35

Check: (x 1)(3x 2) 3x2 2x 3x 2 3x2 x 2

S40

Chapter R

You Try It 11

4a3 4a2 24a There is a common factor, 4a. Factor out the GCF. 4a3 4a2 24a 4a(a2 a 6) Factor a2 a 6. The two factors of 6 whose sum is 1 are 2 and 3. 4a(a2 a 6) 4a(a 2)(a 3) 4a3 4a2 24a 4a(a 2)(a 3)


Check: 4a(a 2)(a 3) (4a2 8a)(a 3) 4a3 12a2 8a2 24a 4a3 4a2 24a

Answers to Chapter 1 Selected Exercises PREP TEST 1. 127.16

2. 55.107

3. 4517

4. 11,396

5. 24

6. 24

8. 3 7

7. 4

2 5

9.

SECTION 1.1 1. 8 6

3. 12 1

17. True

5. 42 19 21. 23, 18

19. False

37. 0

41. 82

39. 74

51. 45 61 69. 5 91. 12

73. 1

25. 23

45. 83 58

55. 45, 0, 17 77. 7

75. 8

95. 9

93. 0

9. 53 46

23. 21, 37

43. 81

53. 19, 0, 28

71. 9

7. 0 31

59. 11 101. 0

115. 42

117. 28

119. 60

121. 253

131. 7

133. 12

135. 6

137. 7

139. 11

153. 11

167. 315

169. 420

173. 2772

171. 2880

61. 5

141. 14

63. 83

65. 46

109. 12

127. 2

145. 16

143. 15

67. 0 89. 41

107. 8

125. 114 161. 240

129. 8 149. 29

147. 0

165. 216

163. 96

177. The difference in elevation is 7046 m.

179. The difference between the highest and lowest elevations is greatest in Asia. student’s score is 93 points.

35. 77

33. 14

87. 10

85. 18

159. 252

175. 0

31. 28

105. 138

103. 2

15. False

49. 68 42

123. 238

157. 105

155. Undefined

13. True

29. 9

83. 3

81. 9

111. 20

151. Undefined

27. 4

47. 43 52

79. 9

99. 18

97. 11

11. False

181. The difference is 5°C.

185. The difference between the average temperatures is 0°F.

183. The

187. The difference is 9°F.

SECTION 1.2 3. 0.2

1. 0.125 21.

22 , 0.88 25

23.

8 , 1.6 5

39. 0.091

41. 0.167

57. 0.4%

59. 83%

77. 0

79.

3 8

9. 0.583

87 , 0.87 100

27.

25.

43. 0.009

7 60

3 8

83.

99.

115. 4.164

117. 4.347

131. 2401

133. 64

149. 144

151. 18

167. 29

169. 542

183. 16.583

7. 0.5625

1 10

4 63. 44 % 9 1 16

4 9

119. 4.028

153. 4

31.

33.

15 64

27 64

107.

175. 482

5 26

10 9

71.

11 8

109.

37.

73.

147 32

1 12

7 24

75.

93. 17.5

111.

25 8

113.

2 3

129. 2060.55

145. 8

147. 6750

163. 1010

179. 16.971

23 400

55. 136%

127. 5.11

161. 182

177. 15.492

1 16

91. 37.19

143. 1

141. 3.375

2 , 0.40 5

53. 12.5%

125. 2.59

159. 22

35.

51. 2% 69.

19.

1 400

89. 10.7893

123. 1.104

157. 42

3 8

49. 37% 67. 250%

105.

15. 0.45

165. 15

181. 16

189a. The average monthly net income was $2.668 million. b. The

187. 18.762

difference is $3325.286 million.

7 30

139.

173. 0

3 70

87. 23.845

121. 2.22

137. 125

171. 35

185. 15.652

65. 45%

103.

155. 7

29.

13. 0.225

47. 0.1823

85. 1.06

101.

135. 8

11. 0.24

9 , 4.50 2

45. 0.0915

1 61. 37 % 2

81.

97.

95. 19.61


5. 0.16

191. The amount spent on decorations is 25% of the total.

193. 3, 4, 5, 6, 7, 8, 9

SECTION 1.3 3. 0 27. 5

5. 11 29. 1

7. 20

9. 10

31. 4

11. 20

33. 0.51

13. 29

15. 11

19. 11

17. 7

21. 6

1 1 1 1 37. Row 1: , 0; Row 2: ; Row 3: , 6 2 3 2

35. 1.7

23. 15

25. 4

39. Your savings on

gasoline would pay for the increased cost of the car in 39 months.

SECTION 1.4 1. 9

3. 41

5. 7

25. 3

27. 2

45. 6ab

47. 12xy

7. 13

29. 4 49. 0

9. 15

31. 10 51.

11. 41

33. 25 1 y 10

53.

13. 1 35. 25x

2 2 y 9

15. 5 37. 9a

55. 20x

17. 1 39. 2y

57. 4a

19. 57

21. 5

41. 12y 3 59. 2y2

23. 8 43. 9a

61. 2x 8y

63. 8x

A1

A2

Chapter 1

65. 19a 12b 81. 108b 103. 6y

67. 12x 2y 56x2

83.

107. 2x

105. 3y

119. 20 14b

85.

141. 2x 3y

1 3

12x

189. x2 2x

1 5 7 x x; x 3 8 24

x 18

171.

1 4 x; x 8 3 3

199. (x 5) 2; x 7

99. 2x

115. 5a 80

127. 14x 49

137. 3x2 6x 18

157. 3y 3

117. 15y 35

129. 30x2 15

159. 2a 4b 173. x 20

149. 2x 16

161. 4x 24

163. 2x 16

175. 10(x 50); 10x 500

15 x 12

14 2 2 (x 7); x 3 3 3

185.

193. x (x 9); 2x 9

201. 2(6x 7); 12x 14

195. x (8 x); 2x 8

203. Let n be the number of nations

205. Let g be the amount of oil in one container; g, 20 g

1 of pecans produced in Texas; p 2 3 217. x 5

101. 15a2

139. 2y2 4y 8

147. 3a2 5a 4

183.

79. 28a

77. 72x

97. 2x

113. 10x 35

181. 4(x 19); 4x 76

191. (x 8)

75. 30y

95. n

145. 6x2 3xy 9y2

169. x 50

participating in 1896; n 1990

93. x

125. 10x 35

1 x 2y 2

155. 2x 41

179. x (x 3); 3

73. 10a

111. x 7

18x2

143. 10x2 15x 35

167. 4x 12

x 20

89. a

135.

133. 5x2 5y2

165. 3x 21

197.

91. b

123.

153. 7n 7

187. 40

87. x 109. 9y

151. 12y 9

5 x6 8

71. 60x

121. 4x 2y

131. 24y2 96

177.

69. 7x2 5x

x2

209. Let d be the diameter of a baseball; 4d

211. Yes

207. Let p be the pounds 213. 2x

215.

1 x 4

SECTION 1.5 1. A {16, 17, 18, 19, 20, 21}

3. A {9, 11, 13, 15, 17}

11. A B {10, 9, 8, 8, 9, 10} 17. A B {4, 5}

19. A B

25. {x x 30, x integers}

5. A {b, c}

13. A B {a, b, c, d, e, f} 21. A B {c, d, e}

39.

29. {x x 8, x real numbers}

33. − 5 −4 −3 −2 − 1

35.

23. {x x 5, x negative integers}

27. {x x 5, x even integers}

31.

9. A B {3, 4, 5, 6}

15. A B {1, 3, 7, 9, 11, 13}

0

1

2

3

4

5

−5 − 4 − 3 −2 −1

0

1

2

3

4

5

−5 −4 −3 −2 −1

0

1

2

3

4

5

37.

− 5 −4 −3 −2 − 1

0

1

2

3

4

5

− 5 −4 −3 −2 − 1

0

1

2

3

4

5

41a. Never true b. Always true

c. Always true

43a. Yes b. Yes

CHAPTER 1 REVIEW EXERCISES* 2. 4 [1.1B]

7. 20 [1.1D]

8. 0.28 [1.2A]

13.

72 85

18. 31 [1.3A]

19. 29 [1.4A]

− 5 − 4 − 3 − 2 −1

is 98. [1.1E]

1

15.

2

3

4

5

16 81

28.

5. 1 [1.1C]

10. 62.5% [1.2B]

[1.2E]

20. 8a 4b [1.4B]

24. 90x 25 [1.4D] [1.5C]

0

4. 13 [1.1C]

9. 0.062 [1.2B]

14. 4.6224 [1.2D]

[1.2D]

23. 7x 46 [1.4D] 27.

3. 5 [1.1B]

16. 12 [1.2F] 21. 36y [1.4C]

25. {1, 3, 5, 7} [1.5A] −5 − 4 −3 −2 − 1

11.

0

1

2

3

4

[1.4E]

[1.2C]

12. 1.068 [1.2C]

17. 630 [1.2F] 22. 10x 35 [1.4D]

26. A B {1, 5, 9} [1.5A] [1.5C]

29. The student’s score

31. 2x

1 3 x; x 2 2

5

30. 50.8% of the candy consumed was chocolate. [1.2G]

32. Let A be the number of American League players’ cards; 5A

7 12

6. 42 [1.1D]

[1.4E]

33. Let T be the number of ten-dollar bills;

35 T [1.4E] *Note: The numbers in brackets following the answers to the Chapter Review Exercises are a reference to the objective that corresponds to that problem. For example, the reference [1.2A] stands for Section 1.2, Objective A. This notation will be used for all Prep Tests, Chapter Review Exercises, Chapter Tests, and Cumulative Review Exercises throughout the text.


1. 4, 0 [1.1A]

A3

Answers to Selected Exercises

CHAPTER 1 TEST 1. 2 40 [1.1A]

8. 0.7 [1.2A]

7. 17 [1.1D]

19.

9.

9 , 0.45 [1.2B] 20

14. 65 [1.2F]

13. 12 [1.2E] 6x2

3. 4 [1.1B]

2. 4 [1.1B]

21y2

23. {x x 3, x real numbers} [1.5B] 25.

[1.5C]

− 5 −4 −3 −2 − 1

0

1

2

3

4

1 15

5. 16 [1.1C]

6. 4 [1.1C] 12.

11. 5.3578 [1.2D]

[1.2C]

16. 22 [1.4A]

17. 5x [1.4B]

21. 7x 33 [1.4D]

1 2

[1.2D]

18. 2x [1.4C]

22. {2, 1, 0, 1, 2, 3} [1.5A]

24. A B {1, 2, 3, 4, 5, 6, 7, 8} [1.5A] 26.

5

27. 10(x 3); 10x 30 [1.4E]

10.

15. 17 [1.3A]

20. x 6 [1.4D]

[1.4D]

4. 14 [1.1C]

−5 − 4 − 3 −2 − 1

[1.5C] 0

1

2

3

4

5

28. Let s be the speed of the catcher’s return throw; 2s [1.4E]

29a. The balance of trade increased from the previous year in 1981, 1988, 1989, 1990, 1991, and 1995. [1.2G] b. The difference was $288.6 billion. [1.2G] c. The difference was greatest between 1999 and 2000. [1.2G] d. The trade balance was approximately 4 times greater in 1990 than in 1980. [1.2G] e. The average trade balance per quarter for the year 2000 was $92.425 billion. [1.2G]

30. The difference is 215.4°F. [1.2G]

Answers to Chapter 2 Selected Exercises PREP TEST 1. 0.09 [1.2B]

2. 75% [1.2B]

7. 1.66x 1.32 10. 5 x

3. 63 [1.4A]

8. 5 2n [1.4E]

[1.4D]

4. 0.65R [1.4B]

5.

7 x 6

6. 9x 18

[1.4B]

[1.4D]

9. Speed of the old card: s; speed of the new card: 5s [1.4E]

[1.4E]

SECTION 2.1 3. Yes

5. No 29. 3

49. 1

1 51. 2

93.

1 3

75. 80

1 2

97.

95.

129. 200%


113. 28 131. 400

11. No

33. 7

3 53. 4

73. 9

111. Equal to room.

9. Yes

31. 0

27. 6

71. 2

7. No

35. 7

1 55. 12

99.

17. Yes

37. 12

39. 5

41. 15

15 7

83. 18

81. 12

101. 4

103. 3

117. 64

119. 24%

135. 200

139. 30

151. Octavia will earn the greater amount of interest. is 50%.

67. 3 89. 0

69. 0 8 3

91.

109. 2.13 125. 9

127. 25%

141. There are 4536 L of oxygen in the 149. Andrea must invest $1875.

153. $1500 was invested at 8%.

157. There are 131.25 lb of wool in the carpet.

161. The percent concentration of the resulting mixture is 6%.

165. Marcella’s average rate of speed is 36 mph.

155. There are 1.8 g of

159. The percent concentration of sugar 163. The runner will travel 3 mi.

167. It would take Palmer 2.5 h to walk the course.

169. The two joggers will meet 40 min after they start. 175. a. Answers will vary.

47. 1

145. 47.9% of the U.S. population watched Super Bowl XXXVIII.

147. You need to know the number of people 3 years old and older in the U.S. platinum in the necklace.

87. 20

123. 400

25. 15

45. 14

63. 9.257

107. 2.06

121. 7.2

23. 2

43. 9

85. 15

105. 4.745

137. 400

143. The median income was $42,428.

19. No

61. 0.283

59. 0.6529

79. 7

115. 0.72 133. 7.7

15. No

7 57. 12

77. 0 3 2

13. Yes

b a

173. x , a 0

171. It will take them 0.5 h.

b. Answers will vary.

SECTION 2.2 1. 3 27. 0

3. 6 3 29. 4

5. 1 4 31. 9

7. 3 1 33. 3

9. 2

11. 2

1 35. 2

13. 5

3 37. 4

15. 3 1 39. 3

17. 6

1 41. 6

19. 3 43. 1

21. 1 45. 1

23. 6 47. 0

25. 7 49.

13 10

A4

Chapter 2

51.

2 5

73.

15 2

53.

4 3

75.

55.

18 5

95. 3

97. 0

115. 41

117. 8

135.

10 3

3 2

57. 18

77. 2

137.

63. 25

83. y 3

101. 3

103. 1

105. 4

121. 4

123. 1

125. 1

119. 1 1 4

81. x 7

79. 3

99. 1

61. 16

59. 8

141. 1

139. 0

85. 2 107.

3 4

65.

5 6

109.

127. 24

16 9

69.

89. 1

87. 3

2 3

3 8

67.

129. 495

93. 2

91. 2

3 4

111.

113. 17 1 2

131.

133.

1 3

143. A force of 25 lb must be applied to the other end.

145. The fulcrum is 6 ft from the 180-pound person.

147. The fulcrum is 10 ft from the 128-pound acrobat.

149. The minimum force to move the rock is 34.6 lb.

151. The break-even point is 260 barbecues.

even point is 520 desk lamps.

1 18

71.

155. The break-even point is 3000 softball bats.

153. The break159. 0

157. No solution

SECTION 2.3 1. x 15 7; 22

3. 7x 21; 3

13. 34x 7 15; 3 23. 4, 6

5. 9 x 7; 2

15. 3x 220 x ; 8, 12

7. 5 2x 1; 2

9. 2x 5 15; 5

17. 2x 14 x 1 ; 5, 9

sides are 6 ft, 6 ft, and 11 ft. to paint the house.

31. The union member worked 168 h during March.

35. There are 1024 vertical pixels.

piece is 3 ft; the longer piece is 9 ft.

21. 1, 1, 3

19. 15, 17, 19

27. The processor speed of the newer personal computer is 4.26 GHz.

25. 5, 7

11. 4x 6 22; 7

29. The lengths of the

33. 37 h of labor was required

37. The length is 13 m; the width is 8 m.

39. The shorter

41. The larger scholarship is $5000.

SECTION 2.4 3. The mixture will cost $1.84 per pound.

7. The amount of olive oil is 2 c; the amount of vinegar is 8 c. 11. To make the mixture, 16 oz of the alloy are needed. 63 lb.

15. There were 228 adult tickets sold.

17. The cost per pound of the sugar-coated cereal is $.70. 21. The amount of the 15% acid is 20 gal.

25. The amount of 9% nitrogen plant food is 6.25 gal.

sugar in the mixture is 19%.

29. 20 lb of 40% java bean coffee must be used.

100 ml; the amount of the 4% solution is 200 ml. concentration of the resulting alloy is 50%. alongside the motorboat.

27. The percent concentration of

33. 150 oz of pure chocolate must be added.

35. The percent

37. The first plane is traveling at a rate of 105 mph; the second plane is 41. In 2 h the cabin cruiser will be

43. The corporate offices are 120 mi from the airport.

47. The distance between the airports is 300 mi. 51. The cyclists will meet after 1.5 h.

19. The

23. The amount of

31. The amount of the 7% solution is

39. The planes will be 3000 km apart at 11 A.M.

traveling at a rate of 130 mph.

must be added.

9. The cost of the mixture is $3.00 per ounce.

13. The amount of almonds is 37 lb; the amount of walnuts is

percent concentration of gold in the mixture is 24%. the 25% wool yarn is 30 lb.

5. The amount of caramel is 3 lb.

45. The rate of the car is 68 mph.

49. The planes will pass each other 2.5 h after the plane leaves Seattle.

53. The bus overtakes the car at 180 mi.

57. 3.75 gal of 20% antifreeze must be drained.

55. 75 g of pure water

59. The bicyclist’s average speed is 13

1 mph. 3

SECTION 2.5 5. x x 5

3. a, c

9. x x 4

1 31. x x 2

45. x x

5 4

21. x x 2

23 33. x x 16

47. x x 2

57. x x 6 or x 4

67. x 2 x 6

23. x x 3

8 35. x x 3

69. x 3 x 2

61. 71.

−5 − 4 − 3 −2 − 1 0 1 2 3 4 5

13. x x 4

25. x x 2

14 39. x x 11

53. x x 3 or x 5 63. x2 x 1

5 x x 5 or x 3

15. x x 2

27. x x 3

37. x x 1

51. x2 x 4

59. x x 3

11. x x 3

− 5 − 4 −3 − 2 −1 0 1 2 3 4 5

19. x x 2

7. x x 2

− 5 −4 −3 − 2 − 1 0 1 2 3 4 5

17. x x 2

29. x x 5 41. x x 1

43. x x

55. x4 x 2

65. x x 2 or x 2 73.

75. The set of real numbers

7 4


1. The amount of $1 herbs is 20 oz.


77.

x

17 45 x 7 7

is 12.

79.

x 5 x

17 3

87. The maximum width of the rectangle is 11 cm.

for more than 460 pages.

85. The smallest number

89. The TopPage plan is less expensive when service is 93. 32 F 86

91. Paying with cash is less expensive when the call is 7 min or less.

95. George’s amount of sales must be $44,000 or more. are written.

83.


99. 58 x 100

103. a. Always true

A5

97. The first account is less expensive if more than 200 checks

101. The three even integers are 10, 12, and 14; or 12, 14, and 16; or 14, 16, and 18.

b. Sometimes true

c. Sometimes true

d. Sometimes true

e. Always true

SECTION 2.6 5. 7 and 7

7. 4 and 4

9. 6 and 6

15. No solution

17. 1 and 5

19. 8 and 2

21. 2

29. No solution

31. 7 and 3

33. 2 and

1. Yes

43. 55.

3. Yes

1 and 1 3

7 1 and 3 3

45. No solution 57.

1 2

1 2

14 x x 2 or x 5

75. 83. 91.

x 2 x

9 2

x x 0 or x

77.

x x 2 or x

89.

27.

11 1 and 6 6

41.

65. x x 3 or x 3

3 2

53. No solution

73. x 3 x 2

81.

22 9

x x

1 or x 3 3

3 2

x x

9 2

95. The lower and upper limits of the diameter of the bushing are

97. The lower and upper limits of the voltage of the electric motor are 195 volts and 245 volts.

99. The lower and upper limits of the length of the piston rod are 9 of the resistor are 28,420 ohms and 29,580 ohms.

19 21 in. and 9 in. 32 32

101. The lower and upper limits

103. The lower and upper limits of the resistor are 23,750 ohms and

105. a. x x 3

26,250 ohms.

13 3

63. No solution

87.

3 and 3 2

39. No solution

51. 1 and

71. x x 5 or x 1

93. x x 5 or x 0

1.742 in. and 1.758 in.

8 10 and 3 3


49. No solution

61.

69. x 4 x 6

85. x x 2 4 5

7 2

3 2

37.

13. No solution

25.

23. No solution

35. 1 and 3

47. 3 and 0

59. and

67. x x 1 or x 3

10 3

11. 7 and 7

b. a a 4

107. a.

b.

c.

d.

e.

CHAPTER 2 REVIEW EXERCISES 1. 6 [2.2B] 6.


11.

40 7

26 17

2. x x

[2.2B] [2.2C]

7.

2 3

5 3

[2.5A]

8. 1 and 9

[2.6A]

12. x x 2 or x 2 [2.5B]

13.

[2.1C]

15.

17 2

[2.2C]

16. No solution [2.6A]

20.

1 12

[2.1B]

21. 7 [2.1C]

26. 8 [2.2C]

27. x x 1

31. x1 x 2 [2.5B] the dock. [2.4C]

4. 9 [2.1B]

3. No [2.1A]

22.

[2.5A] 5 11 2 2

32. ,

2 3

5.

x 3 x

9. x 1 x 4 [2.6B]

x x 2 or x

1 2

[2.1C]

23.

28. x x 4 [2.6A]

8 5

[2.2B]

[2.5A]

33.

8 5

[2.6A]

38. 5x 4 16; x 4 [2.3A]

the bushing are 2.747 in. and 2.753 in. [2.6C]

18.

39 2

34. [2.6B]

5 2

13 11 are 10 in. and 10 in. 32 32

[2.6C]

[2.5B] 9 19

[2.2C]

[2.2C]

19. 15 [2.1B]

25. 9 [2.2C] [2.5A]

30. x x 1 [2.5A]

35. The island is 16 mi from

37. The executive’s amount of sales

39. The lower and upper limits of the diameter of

40. The integers are 6 and 14. [2.3A]

41. 82 x 100 [2.5C]

42. The speed of the first plane is 520 mph. The speed of the second plane is 440 mph. [2.4C] tin alloy and 125 lb of the 70% tin alloy were used. [2.4B]

14. 250% [2.1D]

24. 3 [2.2B]

29. x x

4 3

10.

[2.6B]

17. The set of real numbers [2.5B]

36. The mixture must contain 52 gal of apple juice. [2.4A]

must be $55,000 or more. [2.5C]

43. 375 lb of the 30%

44. The lower and upper limits of the length of the piston

A6

Chapter 3

CHAPTER 2 TEST 1 8

1. 2 [2.1B]

2.

8. No [2.1A]

9.

13. [2.5B]

14.

17.

xx

12 7

1 or x 2 2

120 mi. [2.5C]

[2.1B]

[2.1C]

4. 4 [2.2A]

10. x x 3 [2.5A]

[2.2C] 9 and 3 5

5 6

3.

[2.6B]

16.

x

6.

[2.2A]

1 5

[2.2B]

7. 1 [2.2C]

12. xx 2 [2.5B]

11. 0.04 [2.1D]

15. 7 and 2 [2.6A]

[2.6A]

32 3

5.

13 x 3

[2.6B]

18. It costs less to rent from Gambelli Agency if the car is driven less than

19. The lower and upper limits of the diameter of the bushing are 2.648 in. and 2.652 in. [2.6C]

20. The integers are 4 and 11. [2.3A]

21. 1.25 gal are needed. [2.4B]

mixture is $2.70 per pound. [2.4A]

22. The price of the hamburger

23. The jogger ran a total distance of 12 mi. [2.4C]

train is 60 mph. The rate of the faster train is 65 mph. [2.4C]

24. The rate of the slower

25. It is necessary to add 100 oz of pure water. [2.4B]

CUMULATIVE REVIEW EXERCISES 3.

2. 48 [1.1D]

1. 6 [1.1C]

19 48

[1.2C]

8. 5a 4b [1.4B]

9. 2x [1.4C]

13. 6x 34 [1.4D]

14. A B {4, 0} [1.5A]

17. 25 [2.1C]

18. 3 [2.2A]

23. {x x 1} [2.5A] 27.

11 20

4. 54 [1.2E]

15.

[1.2B] 28. 103% [1.2B]

12. 4x 14 [1.4D] 16. Yes [2.1A]

1 [2.2B] 2

22. 13 [2.2C]

4 3

26. x x 2 or x

25. 1 and 4 [2.6A]

29. 25% of 120 is 30. [2.1D]

31. 20 lb of oat flour must be used. [2.4A]

21.

7. 17x [1.4B]

6. 6 [1.4A]

[1.5C]

− 5 − 4 − 3 − 2 −1 0 1 2 3 4 5

20. 3 [2.2B]

24. {x4 x 1} [2.5B]

[1.3A]

11. 2x2 6x 4 [1.4D]

10. 36y [1.4C]

19. 3 [2.2A]

49 40

5.

[2.6B]

30. 6x 13 3x 5; x 6 [2.3A]

32. 25 g of pure gold must be added. [2.4B]

33. The length of the

track is 120 m. [2.4C]

Answers to Chapter 3 Selected Exercises PREP TEST 1. 43 [2.1B]

2. 51 [2.1B]

3. 56 [1.3A]

4. 56.52 [1.4A]

5. 113.04 [1.4A]

6. 120 [1.4A]

1. 40°; acute

3. 115°; obtuse

5. 90°; right

11. The length of BC is 14 cm. of MON is 86°. 35. 141°

19. 71°

37. 106°

21. 30°

39. 11°

49. x 155°, y 70°

7. The complement is 28°.

13. The length of QS is 28 ft. 23. 36°

25. 127°

41. a 38°, b 142°

51. a 45°, b 135°

57. The measure of the third angle is 35°.

9. The supplement is 18°.

15. The length of EG is 30 m. 27. 116°

29. 20°

31. 20°

43. a 47°, b 133°

53. 90° x

17. The measure 33. 20°

45. 20°

47. 47°



61. a. 1° b. 179°

65. 360°

SECTION 3.2 1. Hexagon 17. 47 mi

3. Pentagon

5. Scalene

7. Equilateral

19. 8 cm or approximately 25.13 cm

approximately 53.41 ft

25. The perimeter is 17.4 cm.

31. The perimeter is 48.8 cm. 37. The circumference is 1.5 in.

9. Obtuse

11. Acute

13. 56 in.

21. 11 mi or approximately 34.56 mi 27. The perimeter is 8 cm.

33. The perimeter is 17.5 in.

15. 14 ft

23. 17 ft or

29. The perimeter is 24 m.

35. The length of a diameter is 8.4 cm.

39. The circumference is 226.19 cm.

41. 60 ft of fencing should be purchased.


SECTION 3.1


43. The carpet must be nailed down along 44 ft. 49. The length of each side is 12 in. 55. The bicycle travels 50.27 ft. 63. 546

65. 16

ft 2

cm2

77. The area is 13.5

51. The length of a diameter is 2.55 cm. 67. 30.25

cm2

71. The area is 156.25 cm2. 79. The area is 330

85. The area is 10,000 in 2.

mi 2

61. 20.25 in 2 69. 72.25 ft 2

or approximately 95.03 mi2

73. The area is 570 in 2. in 2.

75. The area is 192 in 2.

83. The area is 9.08 ft 2.

89. 7500 yd 2 must be purchased.

95. You should buy 2 qt.

97. It will cost $98.

103. The area is 216 m2.

101. The cost will be $878.

53. The length is 13.19 ft.

59. 60 ft 2

81. The area is 25

cm2.

87. The area is 126 ft 2.

93. The length of the base is 20 m. is 113.10 in 2.

47. The length of the third side is 10 in.

57. The circumference is 39,935.93 km.

or approximately 50.27

or approximately 226.98 ft2 ft 2.

45. The length is 120 ft.

A7

91. The width is 10 in.

99. The increase in area 109. A

105. The cost is $1600.

d2 4

SECTION 3.3 1. 840 in 3

5. 4.5 cm3 or approximately 14.14 cm3

3. 15 ft 3

42.875 in 3.

11. The volume is 36 ft 3.

17. The volume is 120 in 3. 301.59 in 2

19. The width is 2.5 ft.

33. The surface area is 184 ft2.

is 3 cm.

21. The radius of the base is 4.00 in. 27. 94 m 2

47. 11 cans of paint should be purchased.

pyramid is 22.53 cm2 larger.

53. a. Always true

49. 456

in2

23. The length is 5 in.

31. 96 in2 or approximately

29. 56 m 2

37. The surface area is 225 cm2.

35. The surface area is 69.36 m2.

41. The surface area is 6 ft2.

39. The surface area is 402.12 in2.

9. The volume is

13. The volume is 8143.01 cm 3. 15. The volume is 75 in3.

25. There are 75.40 m 3 in the tank.

The width is 5 in.

7. The volume is 34 m 3.

43. The surface area is 297 in2. of glass are needed.

45. The width

51. The surface area of the

b. Never true c. Sometimes true

CHAPTER 3 REVIEW EXERCISES 1. x 22°, y 158°

[3.1C]

5. The volume is 96 cm3. [3.3A]

2. x 68° [3.1B]

is 8 cm. [3.2B]

measures 21.5 cm. [3.2A] needed. [3.2A]

10. The area is 63 m2. [3.2B]

12. The measure of the third angle is 95°. [3.1C]

14. The volume is 288 mm3. [3.3A]

15. The volume is

19. The area is 90.25

[3.2B]

20. The area is 276

13. The length of the base

784 cm3. [3.3A] 3

17. 4 cans of paint should be purchased. [3.3B] m2.

4. 19° [3.1A]

7. The surface area is 220 ft2. [3.3B]

9. The area is 78 cm2. [3.2B]

8. The supplement is 148°. [3.1A] 11. The volume is 39 ft3. [3.3A]

3. The length of AC is 44 cm. [3.1A]

6. a 138°, b 42° [3.1B]

16. Each side

18. 208 yd of fencing are

m2.

[3.2B]

CHAPTER 3 TEST


1. The radius is 0.75 m. [3.3A] 4. BC 3 [3.1A]

2. The circumference is 31.42 cm. [3.2A]

5. The volume is 268.08

7. a 100°, b 80° [3.1B] 11. The volume is 169.65 is 164.93 ft2. [3.3B]

ft 3.

[3.3A]

6. The area is 63.62 cm 2. [3.2B]

9. a 135°, b 45° [3.1B]

8. 75° [3.1A]

m3.

[3.3A]

3. The perimeter is 26 ft. [3.2A]

12. The perimeter is 6.8 m. [3.2A]

10. The area is 55 m2. [3.2B] 13. 58° [3.1A]

15. There are 113.10 in 2 more in the larger pizza. [3.2B]

angles are 58° and 90°. [3.1C]

18. The area of the room is 28 yd2. [3.2B]

17. The bicycle travels 73.3 ft. [3.2A]

19. The volume of the silo is 1145.11 ft 3. [3.3A]

14. The surface area

16. The measures of the other two

20. The area is 11 m2. [3.2B]

CUMULATIVE REVIEW EXERCISES 1. 3, 0, and 1 [1.1A] [1.2F]

2. 0.089 [1.2B]

7. 28 [1.3A]

12. {2, 1} [1.5A]

8. 8 [1.4A]

3. 35% [1.2B]

4.

9. 3m 3n [1.4B]

13. {10, 0, 10, 20, 30} [1.5A]

14.

2 3

[1.2D]

5. 24.51 [1.2D]

10. 21y [1.4C]

−5 − 4 −3 −2 −1 0 1 2 3 4 5

6. 55

11. 7x 9 [1.4D] [1.5C]

15. 5 [2.2B]

A8

16.

Chapter 4

1 2

17. { y y 4} [2.5A]

[2.2C]

21. 1,

20. {x 2 x 6} [2.5B]

24. 4x 10 2; x 3 [2.3B]

1 3

18. {x x 2} [2.5A]

19. {x x 3 or x 4} [2.5B]

22. {x 6 x 10} [2.6B]

[2.6A]

25. The third angle measures 122°. [3.1C]

two accounts in one year. [2.1D]

23. x 131° [3.1B] 26. Michael will earn $312.50 from the

27. The third side measures 4.5 m. [3.2A]

28. The women’s median annual

29. The area is 20.25 cm2. [3.2B]

earnings are 70.8% of the men’s median annual earnings. [1.2G]

30. The

height of the box is 3 ft. [3.3A]

Answers to Chapter 4 Selected Exercises PREP TEST 1. 4x 12 [1.4D] 7. 1 [1.4A]

3. 2 [1.3A]

2. 10 [1.2F]

4. 11 [1.4A]

5. 2.5 [1.4A]

6. 5 [1.4A]

8. 4 [2.2A]

SECTION 4.1 y

–4

–2

3.

4

4

2

2

0

2

5.

y

4 2

x

4

–4

–2

2

0

x

4

–4

–2

0

–2

–2

–2

–4

–4

–4

9. A2, 5 , B3, 4 , C0, 0 , D3, 2 23. (3, 7)

25. (6, 3)

11. a. 2, 4

27. (0, 1)

7. A2, 3 , B4, 0 , C4, 1 , D2, 2

y

2

b. 1, 3

29. (5, 0)

x

4

15. Yes

31.

17. No

19. No

33.

y 4

4

(2, 4)

2

2

(0, 0) –4

–2

0

(−1, −2)– 2

(−2, − 4)

37.

y 4 2 –4

–2

0

(–3, –1) – 2 –4

(0, 2)

(–2, 0)

x

–4

–2

(–4, –2)

–4

0

2

4

–4

4 4

x

–2

39. a. After 20 min., the temperature is 280F.

y

(0, 1) 2

4

(3, 5)

8

(3, 3)

x

–8

–4

0

4

8

x

–4 –8

b. After 50 min., the temperature is 160F.

41.

Income (billions of $)

35.

2

21. No

y

43.

3.0

y 4 2

2.5 –4

–2

0 –2

2.0 65 70 75 80 85 90

Profit (thousands of $)

–4

2

4

x


1.


A9

SECTION 4.2 3. Yes

5. Yes

7. No

9. Function

b. y $34.75

19. f3 11

31. F4 24

33. F3 4

43. 4h

11. Function

21. f0 4

23. G0 4

35. H1 1

45. a. $4.75 per game

13. Function

37. Ht

b. $4.00 per game

15. Not a function

25. G2 10

3t t2

27. q3 5

29. q2 0

41. sa a3 3a 4

39. s1 6

47. a. $3000

17. a. Yes

49. Domain 1, 2, 3, 4, 5

b. $950

Range 1, 4, 7, 10, 13 51. Domain 0, 2, 4, 6

53. Domain 1, 3, 5, 7, 9

Range 1, 2, 3, 4

55. Domain 2, 1, 0, 1, 2

Range 0

57. Domain 2, 1, 0, 1, 2

Range 0, 1, 2

61. 8

59. 1

63. None

65. None

67. 0

69. None

71. None

Range 3, 3, 6, 7, 9 73. 2

77. Range 3, 1, 5, 9

75. None

83. Range 2, 14, 26, 42

79. Range 23, 13, 8, 3, 7

5 5, , 5 3

85. Range

93. a. 2, 8 , 1, 1 , 0, 0 , 1, 1 , 2, 8

87. Range

99. a. 68F

b. 30 fts

81. Range 0, 1, 4

89. Range 38, 8, 2

b. Yes, the set of ordered pairs defines a function because each member

of the domain is assigned to exactly one member of the range. 97. a. 22.5 fts

1 1 1, , , 1 2 3

95. The power produced will be 50.625 watts.

b. 52F

SECTION 4.3 1.

3.

y

2 –4

–2

2

x

4

–4

–2

7.

y

y

4

2

0

4

2

0

2

x

4

–4

–2

2

0

2

4

x

–4

–2

–2

–2

–2

–4

–4

–4

–4

11.

y

0

2

x

4

–4

–2

15.

y

2

0

4

–4

–2

2

0

2

4

x

–4

–2

–2

–4

–4

–4

19. x-intercept: 4, 0 ; y-intercept: 0, 2

21. x-intercept:

2 –2

y 2

0

4

–4

–4

23. x-intercept:

–2

0

2 2

4

x

–4

–2

–2 –4

25. x-intercept:

y

4 , 0 ; y-intercept: 0, 2 3

4

2 0

y

4

–2

2 2

4

0

–4

3 , 0 ; y-intercept: 0, 3 2

x

–2

–4

–2

0

–2

–2

–4

–4

2

4

x

4

x

9 , 0 ; y-intercept: 0, 3 2

4

2

–2

y

4

x

2

0

–2

4


–2

–4

y

x

y

2

x

4

4

–2

17.

–4

2

4

2

2 –2

13.

y 4

4

–4

0

–2

9.

–4

5.

y 4

4

2

4

x

A10

Chapter 1

27. x-intercept:

3 , 0 ; y-intercept: 0, 2 2

29. Marlys receives $165 for tutoring 15 h.

y

W Wages (in dollars)

4 2 –4

–2

0

2

4

x

–2 –4

200

(15, 165)

100

0

10

t

20

Hours tutoring

31. The cost of receiving 32 messages is $14.40. Cost (in dollars)

C 16 14

(32, 14.40)

12 10 8 10

20

30

t

40

Number of messages

33. The cost of manufacturing 6000 compact discs is $33,000. Cost (in dollars)

C 60,000 50,000 40,000 30,000 20,000 (0, 3000) 10,000 0

2000

(10,000, 53,000) (6000, 33,000)

4000

6000

8000

n

10,000

Number of discs

39.

y 4 2 –4

–2

0

2

4

x

–2 –4

SECTION 4.4 1 3

3.

5.

2 3

7.

3 4

9. Undefined

11.

7 5

15.

13. 0

1 2

21. m 5. The temperature of the oven decreases 5°min.

The average speed of the motorist is 40 mph.

23. m 0.05. For each mile the car is driven, approximately 0.05 gallon of fuel is used. speed of the runner was 343.9 mmin. 33.

–4

–2

37.

y

25. m 343.9. The average

31. 4, 0, 4, 0, 0 39.

y

y

4

4

4

2

2

2

2

0

2

4

x

–4

–2

0

2

4

x

–4

–2

0

2

4

x

–4

–2

0

–2

–2

–2

–2

–4

–4

–4

–4

43.

y

–2

29. 3, 5, 3, 0, 5

4

41.

–4

27. No

35.

y

45.

y

47.

y

4

4

4

2

2

2

2

2

4

x

–4

–2

0

2

4

x

–4

–2

0

2

4

x

2

4

2

4

x

y

4

0

19. m 40.

17. Undefined

–4

–2

0

–2

–2

–2

–2

–4

–4

–4

–4

x


1. 1

A11


49.

51. Increases by 2

y

53. Increases by 2

55. Decreases by

4

2 3

2 –4

–2

0

2

4

x

–2 –4

61. k 1

57. i. D; ii. C; iii. B; iv. F; v. E; vi. A

63. k

13 2

SECTION 4.5 3. y 2x 5 15. y

5. y

2 7 x 3 3

17. y

2 5

3 5

27. y x 39. y x 2

7. y

1 x 2

29. x 3

4 3

75. a. y 1200x, 0 x 26

5 3

2 5 x 3 3

8 3

2 3

59. y

1 x1 2

4 7 x 3 3

71. y

3 2

49. y x 3 61. y 4

73. y x 3 77. a. y 85x 30,000

b. The height of the plane 11 min after takeoff is 13,200 ft. 79. a. y 0.032x 16, 0 x 500

81. a. y 20x 230,000

driven, 11.2 gal are left in the tank.

7 27 x 5 5

37. x 5

3 1 x 2 2

47. y

25 3

69. y x 1

25. y

35. y 2x 3

1 10 x 3 3

45. y

13. y 3x 4

23. y x 3

33. y 3

57. y x

b. It will cost $183,000 to build an 1800-ft2 house. 83. a. y 63x

2 3

43. y

67. x 2

11. y 3x 9

21. y x 7 15 2

55. y x 1

65. y x

5 3

9. y x 5

5 4

31. y x

53. y x 1

3 x 4

5 21 x 4 4

19. y 3x 9

41. y 2x 3

51. y 1 63. y

1 x2 2

b. After 150 mi are

b. 60,000 cars would be sold at $8500 each. 85. f x x 3

b. There are 315 Calories in a 5-ounce serving.

89. a. 10

87. 0

b. 6

95. Answers will vary. The possible answers include 0, 4 , 3, 2 , and 9, 2 .

SECTION 4.6 3. 5 23. y

5.

1 4

7. Yes

2 8 x 3 3

25. y

9. No

11. No

1 1 x 3 3

13. Yes 5 3

27. y x

14 3

15. Yes

17. Yes

29. y 2x 15

19. No 31.

21. Yes

A1 A 2 B1 B2

33. Any equation of

3 2

the form y 2x b , where b 13, or of the form y x c, where c 8.


SECTION 4.7 3. Yes

5. No

7.

9.

y 4

4

2

2

–4 –2 0 –2

2

x

4

–4 –2 0 –2

15.

y

2 2

x

4

–4 –2 0 –2

2

x

4

–4

17.

y

y 4

–4

–4

13.

11.

y

19.

y

y

4

4

4

4

2

2

2

2

–4 –2 0 –2 –4

2

4

x

–4 –2 0 –2 –4

2

4

x

–4 –2 0 –2 –4

2

4

x

–4 –2 0 –2 –4

2

4

x

A12

Chapter 4

21.

23.

y

y

4

4

2

2

–4 –2 0 –2

2

4

x

–4 –2 0 –2

–4

2

x

4

–4

CHAPTER 4 REVIEW EXERCISES 1. 4, 2 [4.1B]

2.

P2 2

[4.2A]

3.

[4.1B]

y

Pa 3a 4

4.

4

4

2

2

–4 –2 0 –2

2

x

4

–4 –2 0 –2

6. Domain 1, 0, 1, 5 Range 0, 2, 4 y 9. x-intercept: 3, 0 y-intercept: 0, 2 4

8. 4 [4.2A]

2

x

4

–4

–4

5. Range 1, 1, 5 [4.2A]

[4.1A]

y

[4.2A]

7. Yes [4.6A]

[4.3C]

2 –4 –2 0 –2

2

4

x

–4

[4.3C]

y

11.

4

4

2

2

–4 –2 0 –2

2

4

x

14. y

12. 2

2

4

x

–4 –2 0 –2

–4

5 23 x 2 2

[4.5A]

15.

[4.1A]

16.

19. y 4x 5 [4.6A]

occupied. [4.5C]

[4.4B]

2 2

4

x

–4 –2 0 –2

–4

18. a. y x 295; 0 x 295

x

4

2

[4.2A]

4

y

4

17. 2, 1, 2

2

–4

y

–4 –2 0 –2

[4.3B]

y 4

–4 –2 0 –2

–4

13. 1 [4.4A]

[4.3A]

y

2

4

x

–4

b. When the rate is $120, 175 rooms will be

5 20. y x 8 2

[4.6A]

21.

[4.3B]

y 4 2 –4 –2 0 –2

2

4

x

–4

22.

[4.3B]

y 4 2 –4 –2 0 –2 –4

2

4

x

2 3

23. y x 1 [4.5A]

24. y

1 x4 4

[4.5B]

25. (4, 0), (0, 6) [4.3C]


10. x-intercept: 2, 0 y-intercept: 0, 3

A13


26. (1, 7) [4.1B]

27.

[4.1C]

y

28.

17

Men

[4.7A]

y

29.

[4.7A]

y

6

6

16 15

–6

14 13

x

–6

0

–6

x

15 16 17 18 19

6

0

x

6

–6

Women

5 3

34.

31. y 2x

[4.5B]

[4.4B]

y

32. y 3x 7

[4.6A]

35. After 4 h, the car has traveled 220 mi.

4 2 –4 –2 0 –2

2

x

4

33. y

[4.6A] Distance (in miles)

7 6

30. y x

3 x2 2

[4.6A] [4.3D]

d 300 200

(4, 220)

100 0 1 2 3 4 5 6 Time (in hours)

–4

t

37. a. y 80x 25,000

36. The slope is 20. The manufacturing cost is $20 per calculator. [4.4A] b. The house will cost $185,000 to build. [4.5C]

CHAPTER 4 TEST 1.

[4.1B]

y

–4

–2

2. 3, 0 [4.1B]

3.

[4.3A]

y

4.

4

4

2

2

2

2

0

x

4

–4

–2

0

2

4

x

–4

–2

–2

–2

–4

–4

–4

[4.5A]

7.

6. No [4.6A]

1 6

8. P2 9

[4.4A]

[4.2A]

9.

2

0

–2

5. x 2

[4.3B]

y

4

4

x

[4.3C]

y 4 2 –4

–2

0

2

4

x

–2 –4

10.

[4.4B]

y

11. y

4

2 x4 5

[4.5A]

12. 0 [4.2A]

7 5

13. y x

1 5

[4.5B]

2 –4

0

–2

2

4

x


–2 –4

14. y 3

[4.5A]

15. Domain 4, 2, 0, 3 [4.2A]

3 2

16. y x

7 2

[4.6A]

Range 0, 2, 5 17. y 2x 1

[4.6A]

18.

[4.7A]

y

19. The slope is 10,000 . The value of the house

4 2 –4 –2 0 –2

2

4

x

–4

decreases by $10,000 each year. [4.4A] enroll. [4.5C]

20. a. y

3 x 175 10

b. When the tuition is $300, 85 students will

A14

Chapter 5

CUMULATIVE REVIEW EXERCISES 1. 5, 3 [1.1A] 6. 4d 9 [1.4B] 10.

3 2

[2.2A]

3. 95

2. 0.85 [1.2B] 7. 32z [1.4C]

11. 1 [2.2C]

15. f(2) 6 [4.2A]

8. 13x 7y

[1.4D]

5 8

[1.4A] [1.5C]

9. −5 −4 −3 −2 −1

12. {x x 1} [2.5A]

16. The slope is 2. [4.4A]

5.

4. 12 [1.3A]

[1.2F]

1

2

3

4

14. x0 x

13. [2.5B]

17.

0

5

10 3

[2.6B]

y 4 2 –4

–2

0

2

x

4

–2

[4.3A]

–4

18.

19.

y

–4

–2

20.

y

y

4

4

4

2

2

2

0

2

x

4

–4

–2

0

–2

2

x

4

–4

–2

–2

[4.3B]

–4

1 21. y x 2 3

[4.5B]

[4.4B]

–4

3 22. y x 7 2

plane is traveling at 200 mph. [2.4C]

0

2

4

x

–2

[4.6A]

[4.7A]

–4

23. The first plane is traveling at 400 mph. The second

24. 20 lb of $6 coffee and 40 lb of $9 coffee should be used. [2.4A]

10,000 25. The slope is . The value of the backhoe decreases by $3333.33 each year. [4.4A] 3

Answers to Chapter 5 Selected Exercises PREP TEST [1.4D]

2. 7 [1.4A]

[4.3A]

y

0

5. 1000 [2.2C]

8. 4

2 2

4

x

−4

−2

0

[4.7A]

y

4

2 −2

[4.3B]

y

4

−4

4. 3 [2.2C]

3. 0 [2.2A]

7.

2 2

4

x

−4

−2

0

–2

–2

–2

–4

–4

–4

2

4

x

SECTION 5.1 1. No

3. Yes

5. Independent

7. Inconsistent

y

9.

−4

−2

y

11.

4

4

2

2

0

2

4

x

−4

−2

0

−2

−2

−4

−4

3, 1

2, 4

2

4

x


1. 6x 5y 6.


y

13.

y

15.

4 2 −4

−2

0

2

4

−4

4

2

0

4

2

x

−4

−2

2

0

2

4

−4

−2

0

−2

−2

−2

−4

−4

−4

−4

2, 1

4, 1 y

−2

y

23.

25. 4

2

2

2

2

4

x

−4

−2

0

2

4

x

−4

−2

0

−2

−2

−2

−4

−4

−4

No solution

27. 2, 1

29. 1, 2

2 , 1 3

31. 3, 2

33.

47. 2, 5

45. No solution

57. The amount invested at 4.2% is $5500.

5 2

, 4

x,

2

2 x2 5

35. 1, 4

49. 0, 0

51.

17. 5, 4

3. 1, 1

x,

37. 2, 5

1 x4 2

39. 1, 5

53. 1, 5

59. The amount invested at 6.5% must be $4000.

55.

41. 2, 0

2 ,3 3

61. The amount

63. There is $6000 invested in the mutual bond

67. 0.54, 1.03

SECTION 5.2 1. 6, 1

x

4

invested at 3.5% is $23,625. The amount invested at 4.5% is $18,375. 65. 1.20, 1.40

x

4

4, 2

4

0

2

y

4

3, 2

fund.

x

−2

21.

43.

−2

y

19.

4

2

x

4, 3

−4

y

17.

4

A15

5. 2, 1

7. 2, 1

19. 2, 5

21.

1 2 2 3

33. 10, 5

35.

,

47. 1, 2, 4

49. 2, 1, 2

61. 0, 2, 0

63. 1, 5, 2

1 3 , 2 4

37.

9. x, 3x 4

23. 0, 0

5 1 , 3 3

65. 2, 1, 1

25. 1, 3

39. No solution

51. 2, 1, 3

11.

1 2

,2

27.

73. y

31. 2, 1

29. (2, 3)

43. 2, 1, 3

55. 1, 4, 1

71. 1, 1

15. 1, 2

13. No solution

2 2 , 3 3

41. 1, 1

53. No solution

69. 1, 1

45. 1, 1, 2

57. 1, 3, 2

59. 1, 1, 3

2x 3x ,y 1 3x x2


SECTION 5.3 1. The rate of the motorboat in calm water is 15 mph. The rate of the current is 3 mph. calm air is 502.5 mph. The rate of the wind is 47.5 mph. the current is 2 kmh.

5. The rate of the team in calm water is 8 kmh. The rate of

7. The rate of the plane in calm air is 180 mph. The rate of the wind is 20 mph.

9. The rate of the plane in calm air is 110 mph. The rate of the wind is 10 mph. flour is $.65. The cost per pound of the rye flour is $.70. $3.30/ft.

15. The cost of the wool carpet is $52/yd.

during the week.

3. The rate of the plane in

11. The cost per pound of the wheat

13. The cost of the pine is $1.10/ft. The cost of the redwood is 17. The company plans to manufacture 25 mountain bikes

19. The chemist should use 480 g of the first alloy and 40 g of the second alloy.

21. The Model V

computer costs $4000.

23. There were 190 regular tickets, 350 member tickets, and 210 student tickets sold for the

Saturday performance.

25. The investor placed $10,400 in the 8% account, $5200 in the 6% account, and $9400 in the

4% account.

27. The measures of the two angles are 35° and 145°.

the age of the watercolor is 50 years.

29. The age of the oil painting is 85 years, and

A16

Chapter 5

SECTION 5.4 y

1.

y

3.

y

5.

4

4

4

2

2

2

−4 −2 0 −2

2

x

4

−4 −2 0 −2

4

y

4 2

−4 −2 0 −2

−4

−4

9.

2

x

2

y 4

4

2

2

4

−4 −2 0 −2

y

13.

2 2

4

−4 −2 0 −2

2

x

4

4 2 2

x

4

−4 −2 0 −2

−4

y

17.

y

19.

y

21.

4

4

2

2

2

−4 −2 0 −2

2

4

−4 −2 0 −2

2

x

4

−4 −2 0 −2

−4

−4

2

x

4

−4

4

x

x

4

y

15.

−4 −2 0 −2

−4

2

−4

4

−4 −2 0 −2

x

−4

11.

x

y

7.

y

23. 4 2 2

4

x

−4 −2 0 −2

−4

2

4

x

−4

CHAPTER 5 REVIEW EXERCISES

6,

1 2

2. 4, 7 [5.2A]

[5.1B]

y

3.

y

4.

4

4

2

2

−4 −2 0 −2

2

x

4

−4 −2 0 −2

5.

10.

5 1 , 2 2

[5.1B]

0, 3 [5.1A]

[5.4A]

12. 3, 2 [5.2A] y

16.

4

x, 2x 4

7. 3, 1, 2 [5.2B]

[5.2A]

11. 3, 1 [5.1B]

[5.1B]

y

15.

6.

1 x, x 2 3

4

x

−4

−4

1 3 x, x 4 2

2

[5.1A]

8. 5, 2, 3 [5.2B]

13. 1, 3, 4 [5.2B]

[5.4A]

2

−4 −2 0 −2

2

x

4

−4 −2 0 −2

−4

2

x

4

−4

17. The rate of the cabin cruiser in calm water is 16 mph. The rate of the current is 4 mph. 18. The rate of the plane in calm air is 175 mph. The rate of the wind is 25 mph. [5.3A] attended. [5.3B]

3, 4 [5.1A]

y 4 2

−4

5, 0 [5.1A]

y

2. 4 2

2

4

[5.3A]

19. On Friday, 100 children

20. The amount invested at 3% is $5000. The amount invested at 7% is $15,000. [5.1C]

CHAPTER 5 TEST

−4 −2 0 −2

14. 2, 3, 5 [5.2B]

4

2

1.

9. Yes [5.1A]

x

−4 −2 0 −2 −4

2

4

x


1.


y

3.

[5.4A]

y

4.

4

[5.1B]

[5.2A]

9. No solution [5.2A]

2

−4 −2 0 −2

2

x

4

−4 −2 0 −2

−4

2

x

4

−4

6. 3, 4

7. 2, 1 [5.1B]

[5.1B]

11. No solution [5.2B]

3 7 , 4 8

5.

4

2

15.

[5.4A]

A17

1 6 3 , , 5 5 5

[5.2B]

8. 2, 1

12. 2, 1, 2 [5.2B]

13.

1 10 , 3 3

[5.1B]

10. 1, 1 [5.2A]

14. Yes [5.1A]

16. The rate of the plane in calm air is 150 mph. The rate of the wind is 25 mph. [5.3A]

17. The cost of cotton is $9/yd. The cost of wool is $14/yd. [5.3B]

18. The amount invested at 2.7% is $9000. The

amount invested at 5.1% is $6000. [5.1C]

CUMULATIVE REVIEW EXERCISES 1. 610 [1.2F]

6. {x 4 x 8} [2.6B] 9.

3. 3x 24 [1.4D]

2. 22 [2.2C]

7. {xx 4 or x 1} [2.6B] 10. y

[1.5C]

−5 −4 −3 −2 −1 0 1 2 3 4 5

3 12. y x 1 [4.6A] 2

13.

5 2 x 3 3

4. 4 [1.4A]

8. F(2) 1 [4.2A] 11. y 5x 11 [4.5B]

[4.5A]

14.

y

y

4

4

2 −4 −2 0 −2

5. {xx 6} [2.5B]

2 2

x

4

−4 −2 0 −2

−4

2

4

x

−4

[4.3B] 15.

16.

y

4

2

2 2

4

17. (1, 0, 1) [5.2B]

y

4

−4 −2 0 −2

x

−4 −2 0 −2

−4

2

4

18. 2, 1, 1 [5.2B]

x

−4

[5.4A]

[5.1A] The solution is (2, 0). 19. (2, 3) [5.2A]

20. (5, 11) [5.1B]

wind is 12.5 mph. [5.3A] Copyright © Houghton Mifflin Company. All rights reserved.

[4.7A]

21. 60 ml of pure water must be used. [2.4A]

23. One pound of steak costs $10. [5.3B]

limit is 13,800 ohms. [2.6C]

22. The rate of the

24. The lower limit is 10,200 ohms. The upper

25. The slope is 40. The slope represents the monthly income (in dollars) per thousand

dollars in sales. [4.4A]

Answers to Chapter 6 Selected Exercises PREP TEST 1. 1 [1.1C]

2. 18 [1.1D]

7. 11x 2y 2 [1.4D]

3.

2 3

8. 0 [1.4B]

[1.1D]

4. 12y [1.4C]

9. No [1.4B]

5. 8 [1.2E]

6. 3a 8b [1.4B]

A18

Chapter 6

SECTION 6.1 3. 18x3y4

1. a4b4

19. 64a24b18

21. x2n1

7. 81x8y12

23. y6n2

9. 729a 10b 6

25. a 2n

2

6n

11. x5y11

37. 54a13b17

55. x5y5

57.

a8 b9

9a 8b6

75.

16x2 y6

59.

91. 1.7 1010

1 2187a

61.

1 b4n

77.

y6 x3

79.

93. 2 1011

103. 20,800,000

41. y3

39. 243

y4 x3

63.

1 y6n

65.

81. yn2

95. 0.000000123

105. 0.000000015

ab 4

43.

1 2x2

47.

67.

1 243a5b10

1 y2n

85.

1 2

xn5 y6

49. 69.

97. 8,200,000,000,000,000

107. 0.0000000000178

8.86 103 h to reach Saturn. 8.64 10 m in 8 h.

8 5

127. a.

1 y8

53.

1 x6y10

3a4 4b3

89. 4.67 106

99. 0.039

109. 140,000,000

33. 6x4y4z5

71.

101. 150,000

111. 11,456,790 117. It takes the satellite

119. A proton is 1.83664508 103 times heavier than an electron.

one revolution in 1.5 107 s.

17. 4096x12y12

51.

16x8 81y4z12

123. The signals from Earth to Mars traveled 1.081 107 mimin.

12

1 9

8b15 3a18

87.

115. It would take a spaceship 2.24 1015 h to cross the galaxy.

113. 0.000008

15. a18b18

31. 12a2b9c2

x y4

45.

83.

13. 729x6

29. 6x5y5z4

27. x15n10 3 2

35. 432a7b11

73.

5. x8y16

121. Light travels

125. The centrifuge makes

5 4

b.

SECTION 6.2 1. P3 13

3. R2 10

11. Not a polynomial

5. f1 11

7. Polynomial: a. 1 b.

13. Polynomial: a. 3


b. 14

c. 0

19.

b. 8

c. 2

21.

y

0

2

4

–4

–2

25. 6x 6x 5

27. x 1

29. 5y 15y 2

31. 7a a 2

2

4

–4

–2

0

2

4

x

–2 –4

33. 3x 3xy 2y

2

39. a. k 8

0

–4

2

37. Sx 3x4 8x2 2x; S2 20

2

x

–2

–4 2

y 4

2

x

–2

2

c. 3 23.

y

2 –2

b. 2

4

4

–4

9. Not a polynomial


c. 5

2

35. 8x 2xy 5y2

2

2

b. k 4

SECTION 6.3 3. 6x4 3x3

5. 6x2y 9xy2

7. xn1 xn

13. 6a 4a 6a

15. 3y 4y 2y

21. x3n x2n xn1

23. a2n1 3an2 2an1

4

3

2

31. x 5x 14

4

71. x4 1

n n

2

73. x2n 9

83. a 2a b b 2n

n n

2n

75. x2 10x 25

85. x 2xy

b. x3 y3

37. x y xy 12

101. a. k 5

39. 2x4 15x2 25

47. 6a2n anbn 2b2n

59. x3n 2x2n 2xn 1 65. 9x2 4

3

77. 9a2 30ab 25b2

87. 4xy

2

2

99. a. a3 b3

4

29. 9b5 9b3 24b

53. 4x5 16x4 15x3 4x2 28x 45

3

5

19. 2x4y 6x3y2 4x2y3

5

2 2

63. 2y 10y y y2 3

2n

93. x 9x 27x 27 cubic centimeters 3

2

57. 2a3 7a2 43a 42

n 2n

11. 4b2 10b

4

27. 6y2 31y

2

45. 6a2n an 15

3

61. x 2x y 4x y 2x y 3y 2n n

25. 5y2 11y

51. 10a 27a b 26ab 12b

2

55. x4 3x3 6x2 29x 21 2n

3

2

43. x2n xn 6

49. x 2x 11x 20

2

35. 8x 8xy 30 y

2

41. 10x4 15x2y 5y2

9. x2n xnyn

17. 20x 15x 15x 20x

2

33. 8y 2y 21

2

3

3

79. x4 6x2 9

89. 3x 10x 8 square feet 2

95. 2x cubic inches 3

b. k 1

67. 36 x2

69. 4a2 9b2

81. 4x4 12x2y2 9y4

91. x 3x square meters 2

97. 78.5x 125.6x 50.24 square inches 2

103. a2 b2

SECTION 6.4 1. x 2

3. x 2

5. xy 2

13. x 8

15. x2 3x 6

23. 3x2 1

1 2x2 5

20 x3

7. x2 3x 5 17. 3x 5

25. x2 3x 10

9. 3b3 4b2 2b

3 2x 1

27. x2 2x 1

19. 5x 7 1 x3

11. a4 6a2 1 2 2x 1

21. 4x2 6x 9

29. 2x3 3x2 x 4

31. 2x

18 2x 3 x2 x2 2x 1


1. 2x2 6x


10x 8 x2 2x 1

33. x2 4x 6 43. 2x2 3x 9

45. 4x2 8x 15 2 x1

51. 3x3 x 4 61. Z3 60

53. 3x 1

63. Q2 31

73. a. a ab b 2

3 2x 1

35. x2 4

4

47. 2x2 3x 7

8 x2

55. P3 8

65. F3 178

3

39. 3x 8

12 x2

b. x x y x y xy y

2

37. 2x 8

2 2

3

8 x4

41. 3x 3

1 x1

49. 3x3 2x2 12x 19

57. R4 43

67. P5 122

A19

33 x2

59. P2 39

69. R3 302

71. Q2 0

c. x x y x y x y xy y

4

5

4

3 2

2 3

4

5

CHAPTER 6 REVIEW EXERCISES 1. 21y2 4y 1 5.

b6 a4

[6.1B]

10. 6a 3b 6

2. 5x 4

[6.2B]

11.

2x3 3

14. 13y3 12y2 5y 1 [6.2B]

21. 16x2 24xy 9y2 [6.3C] 24. 70xy2z6

25.

[6.1B]

29. 4x4 2x2 5 [6.4A]

35. 16u v

30. 2x 3

[6.1A]

[6.1B]

52. 25a2 4b2 [6.3C]

27. 2 10 6

37. 2x 3x 8 [6.2B]

28. 68 [6.4D]

43.

46. 1.27 10 7 [6.1C]

[6.4B/6.4C] 50.

38. a2 49 [6.3C]

42. 2y 9 [6.4B]

x4y6 9

1 16

[6.1B]

47. 4y 8 [6.4A]

51. 54a13b5c7 [6.1A]

[6.1B]

54. 8a3b3 4a2b4 6ab5 [6.3A]

53. 0.00254 [6.1C]

[6.1C]

34. 6y3 17y2 2y 21 [6.3B]

2

41. 108x18 [6.1A]

49. 6x3 29x2 14x 24 [6.3B]

20. x4y8z4 [6.1A]

23. 2x2 5x 2 [6.2B]

[6.4B/6.4C]

33. 8x3 14x2 18x [6.3A]

1 x3

[6.1B]

31. a3n3 5a2n4 2a2n3 [6.3A]

2

45. x 2

1 2a

16. 12x5y3 8x3y2 28x2y4 [6.3A]

19. 33x2 24x [6.3A]

[6.1C]

[6.4B]

40. 14,600,000 [6.1C]

44. 10a2 31a 63 [6.3B] c10 2b17

4 6x 1

36. 2x 9x 3x 12 [6.2B] 3

39. 100a15b13 [6.1A]

48.

26. 9.48 108

[6.1B]

9.

13. 7 [6.2A]

[6.4B/6.4C]

[6.4B/6.4C]

252 x4

22. x3 4x2 16x 64

x3 4y2z3

50 x6

18. 8b2 2b 15 [6.3B]

32. x4 3x3 23x2 29x 6 [6.3B] 12 16

7 b7

15. b2 5b 2

17. 2ax 4ay bx 2by [6.3B]

8. 4b4 12b2 1 [6.4A]

12. 4x2 3x 8

[6.1B]

4. 25y2 70y 49 [6.3C]

3. 144x2y10z14 [6.1B]

[6.4B]

7. 4x2 8xy 5y2 [6.2B]

6. 1 [6.4D] [6.1A]

6 3x 2

55.

[6.2A]

y 4 2 –4

–2

0

2

4

x

–2


–4

56. a. 3

b. 8

c. 5 [6.2A]

57. The mass of the moon is 8.103 1019 tons. [6.1D]

9x2 12x 4 square inches. [6.3D]

58. The area is

59. The Great Galaxy of Andromeda is 1.291224 1019 mi from Earth. [6.1D]

60. The area is 10x2 29x 21 square centimeters. [6.3D]

CHAPTER 6 TEST 1. 4x3 6x2

[6.3A]

2. 8 [6.4D]

6. x3 7x2 17x 15 [6.3B] 11. x 7 [6.4B/6.4C] 14. 16y 2 9 [6.3C]

3.

7. 8a6b3

4 x6

[6.1A]

12. 6y 4 9y 3 18y 2

4. 6x 3 y 6

[6.1B] 8.

[6.3A]

15. 6x3 3x2 2x [6.4A]

9y10 x10

[6.1B]

5. x 1

[6.1A]

2 x1

9. a2 3ab 10b2

[6.3B]

13. 4x4 8x 3 3x2 14x 21 [6.3B]

16. 8ab4 [6.1B]

17.

2b7 a10

[6.1B]

[6.4B/6.4C] 10. 3 [6.2A]

A20

Chapter 7

18. 5a3 3a2 4a 3 [6.2B]

19. 4x2 20x 25 [6.3C]

21. 10x 2 43xy 28y 2

22. 3x 3 6x 2 8x 3 [6.2B]

[6.3B]

20. x2 5x 10 23. 3.02 109

23 x3

[6.4B/6.4C]

[6.1C]

24. There are

25. The area of the circle is ( x 2 10 x 25 ) square meters. [6.3D]

6.048 10 s in 10 weeks. [6.1D] 6

CUMULATIVE REVIEW EXERCISES 1. 3 and 3 [1.1A]

2. 83 [1.1B]

of Addition [1.4B]

7. 186x 8 [1.4D]

8.

1 6

[4.4A]

11. 18 [4.2A]

13.

12. Yes [4.2A]

9 2 11 17. , , 7 7 7

16. 1, 2 [5.1B] 20. 5.01 106 [6.1C]

21.

4.

3. 6 [1.3A]

1 6

–2

[2.2B]

10. 1 and

3 1 14. y x 2 2

[4.5A]

15. y

22. 4

2

2 2

x

4

–4

–2

0

24.

[4.6A]

x

4

[4.7A]

b5 25. 8 a

y

[6.1B]

26.

y2 25x 6

[6.1B]

4

2 –2

2

–4

[4.3B]

4

–4

2 16 x 3 3

–2

–4

y

[2.6A]

y

4

0

7 3

19. 4x3 7x 3 [6.3B]

18. 5x 3xy [1.4B]

[5.2B]

6. The Inverse Property

11 4

9.

–2

23.

5. 503 [1.2F]

[1.4A]

[2.2A]

y

–4

5 4

2

0

2

4

x

–2 0 –2

–4

–2

2

4

x

–4

–4

[5.1A]

[5.4A]

The solution is (1, 1). 27. The two integers are 9 and 15. [2.3A] traveling at 5 mph and 7.5 mph. [2.4C]

28. 40 oz of pure gold must be used. [2.4A]

29. The cyclists are

30. $2000 is invested in the 4% account. [5.1C]

slope represents the average speed in miles per hour. [4.4A]

31. The slope is 50. The

32. The length is 15 m. The width is 6 m. [3.2A]

33. The area is (4x2 12x 9) square meters. [6.3D]

PREP TEST 1. 2 3 5 6.

1 2

2. 12y 15

[1.2D]

7. x 2 2x 24

[2.2A]

[1.4D]

[6.3B]

3. a b

[1.4D]

8. 6x 2 11x 10

4. 3a 3b 9. x 3

[6.3B]

[1.4D]

[6.1B]

5. 0 [2.1C] 10. 3x 3y

[6.1B]

SECTION 7.1 3. 5a 1 17. 2x x3 2

5. 82 a2

7. 42x 3

19. 2x25x2 6

29. 5y y2 4y 1

9. 65a 1

21. 4a52a3 1

31. 3y2 y2 3y 2

11. x7x 3

23. xy xy 1

33. 3y y2 3y 8

13. a23 5a3 25. 3xy xy3 2

35. a26a3 3a 2

15. y14y 11 27. xy x y2

37. ab2a 5ab 7b

39. 2b2b4 3b2 6

41. x28y2 4y 1

43. a z y 7

45. a b 3r s

47. m 7 t 7

49. 4a b 2y 1

51. x 2 x 2y

53. p 2 p 3r

55. a 6 b 4

57. 2z 1 z y


Answers to Chapter 7 Selected Exercises


59. 2v 3y 4v 7 69. a. 28

61. 2x 5 x 3y 71. a. r 2

b. 496

63. y 2 3y a

b. 2r 4

2

65. 3x y y 1

A21

67. 3s t t 2

c. r 4

2

2

SECTION 7.2 3. x 1 x 2

1. The same

13. b 3 b 3

5. x 1 x 2

15. b 8 b 1

7. a 4 a 3

17. y 11 y 5

9. a 1 a 2

19. y 2 y 3

23. z 8 z 20

25. p 3 p 9

27. x 10 x 10

33. b 4 b 5

35. y 3 y 17

37. p 3 p 7

11. a 2 a 1

21. z 5 z 9

29. b 4 b 5

31. x 3 x 14

39. Nonfactorable over the integers

41. x 5 x 15

43. p 3 p 21

45. x 2 x 19

51. a 3 a 18

53. z 21 z 7

55. c 12 c 15

57. p 9 p 15

59. c 2 c 9

61. x 15 x 5

63. x 25 x 4

65. b 4 b 18

67. a 45 a 3

69. b 7 b 18

71. z 12 z 12

73. x 4 x 25

81. ab 8 b 1

83. x y 3 y 5

91. 6z 5 z 3

93. 3aa 3 a 6

109. x 2 x 1 x 12

75. x 16 x 7

103. z z 10 z 8

117. a 6b a 7b

119. y z y 7z

125. 4zz 11 z 3

127. 4x x 3 x 1

133. t 5s t 7s

143. 6, 10, 12

105. b b 2 b 5

113. 3x x 3 x 9

99. s 8t s 6t

107. 3y y 3 y 15 2

115. x 3y x 5y

123. 3x x 4 x 3

129. 5 p 12 p 7

131. p2 p 12 p 3

137. y x 6 x 9

145. 6, 10, 12

89. 2x x2 x 2

97. a 5b a 10b 2

121. 3y x 21 x 1

135. a 3b a 11b

141. 22, 22, 10, 10

79. x 2 x 6

87. 4y y 6 y 3

95. x 7y x 3y

111. 3y x 3 x 5

49. a 4 a 11

77. 3 x 2 x 3

85. 2aa 1 a 2 2


47. x 9 x 4

139. 36, 36, 12, 12

147. 4, 6

SECTION 7.3 1. x 1 2x 1

3. y 3 2y 1

11. x 3 2x 1 the integers

21. 2t 1 3t 4

27. 3y 1 4y 5

23. x 4 8x 1

39. 5 y 1 3y 7

71. 2x 3 3x 4

83. 6t 5 3t 1

91. 3b 5 11b 7


43. ba 4 3a 4

75. 3a 8 2a 3

77. z 2 4z 3

79. 2p 5 11p 2

87. 3x 2 3x 2

89. 2b 3 3b 2

95. 3a 7 5a 3

103. 2z 5 5z 2

97. 2y 5 4y 3

105. 6z 5 6z 7

111. y 2z 4y 3z

113. z 7 z 4

117. 3x 5 3x 4

119. 42x 3 3x 2

121. a25a 2 7a 1

137. 4y 3 y 3

127. 38y 1 9y 1 139. 5, 5, 1, 1

69. xy3x 4y 3x 4y

85. b 12 6b 1

109. a 2b 3a b 125. x 7y 3x 5y

45. Nonfactorable 53. y2x 5 3x 2

61. 2y y 4 5y 2

67. y3x 5y 3x 5y

93. 3y 4 6y 5


35. p 8 3p 2

51. z2z 3 4z 1

59. 2z 4 13z 3

73. b 7 5b 2

81. y 1 8y 9

33. z 14 2z 1

41. x x 5 2x 1

65. 3a2a 3 7a 3

19. Nonfactorable over


49. 44y 1 5y 1

57. p p 5 3p 1

63. yzz 2 4z 3

9. x 1 2x 1

17. 3y 1 4y 1

31. b 4 3b 4

47. 3x x 4 x 3

55. 5t 2 2t 5

7. b 5 2b 1

15. p 5 3p 1

29. a 7 7a 2

37. 2x 1 2x 1 over the integers

5. a 1 2a 1

13. t 2 2t 5

115. x 1 x 8

129. x 1 x 21

141. 5, 5, 1, 1

99. 2z 3 4z 5

107. x y 3x 2y

123. 5b 7 3b 2 133. xx 1

135. 2y 1 y 3

143. 9, 9, 3, 3

SECTION 7.4 1. 4; 25x6; 100x4y4 13. 1 3a 1 3a 21. an 1 an 1

3. 4z4

5. 9a2b3

15. xy 10 xy 10 23. x 6 2

31. Nonfactorable over the integers 41. x y a b x y a b 51. 2x 1 4x2 2x 1 2

9. 2x 1 2x 1

27. 4x 5 2

33. x 3y 2 9

43. 8; x ; 27c d

3

45. 2x

47. 4a2b6

39. x 7 x 1

49. x 3 x2 3x 9 57. 4x 1 16x2 4x 1


67. a 2b a ab b 2

37. xn 3 2

55. m n m 2 mn n 2

2 2

19. 5 ab 5 ab


35. 5a 4b 2 15 18

61. xy 4 x y 4xy 16


11. 4x 11 4x 11


25. b 1 2

53. x y x2 xy y2

59. 3x 2y 9x 6xy 4y 2

7. x 4 x 4

2

69. x2n yn x4n x2nyn y2n

A22

Chapter 7

71. xn 2 x2n 2xn 4

73. xy 5 xy 3

79. b 5 b 18

2 2

2

87. 2ab 3 3ab 7

2 2

n

89. x2 1 2x2 15

91. 2xn 1 xn 3

99. 52x 1 2x 1

105. 2a2 a 4 2a a2

107. b3ab 1 a2b2 ab 1

113. x y x x y y 2

2

4

2 2

4

127. x x 1 n

103. 4x 9 2x 3 2x 3

101. y y 11 y 5

121. 2b23a 5b 4a 9b

95. 32x 3 2

93. 2an 5 3an 2

3

2

111. x2 y2 x y x y

109. 2x22x 5 2

117. 2a2a 1 4a2 2a 1


119. a2b2a 4b a 12b

85. 3xy 5 xy 3

n

97. a3a 1 9a 3a 1 2

77. x2 3 x2 6

75. xy 12 xy 5 83. x 1 x 2

81. x y 2 x y 6

2

123. x 2 2 x 2

125. 2x 1 2x 1 x y x y

129. b b 2 3b 2

2

n

SECTION 7.5 3. 3, 2

5. 7, 3

1 1 3 3

21. , 37. 2, 3

23. 2, 4 39.

55. 8, 9 and 5. league.

7. 0, 5

1 , 4 2

1 ,4 3

1 3

15. 2, 5

45. 2, 9

1 2

31. 0, 3

47. 1, 2

73. The object will hit the ground 3 s later.

7 7 2 2

17. 9, 9

19. ,

33. 0, 7

35. 1, 4

49. 9, 5

51. 7, 4

63. The numbers are 2 and 4.

69. There will be 12 consecutive numbers.

77. The length is 15 in. The width is 5 in. 3 87. , 5 2

2 3

29. ,

61. The number is 6.

67. The numbers are 3 and 7.

area are 4 in. by 7 in.

13. 0,

27. , 5

43. 3, 9

59. 5, 2

57. 1, 4

3 2

1 2

25. 7, 2 41.

11. 0,

9. 0, 9

53. 2, 3

65. The numbers are 4

71. There are six teams in the

75. The golf ball will return to the ground 3.75 s later.

79. The height of the triangle is 14 m.

81. The dimensions of the type

83. The radius of the original circular lawn was approximately 3.81 ft.

85. 3, 48

89. 0, 9

CHAPTER 7 REVIEW EXERCISES 1. b 10 b 3 [7.2A] 4. 7x 2y2 3 3x 2y2 2

2. x 3 4x 5 [7.1B]

7. 2x 7 3x 4 [7.3A]

8. 3ab4a b [7.1A]

11. 6y 1 2y 3 [7.3A] 14. c 6 c 2 [7.2A]

6. y 9 y 4 [7.2A]

9. a 10 a3 10 [7.4A]

15. 6a 5 3a 2 [7.3B]

21. 3a 5b 7x 2y [7.1B] 24. 5 x 3 x 2 [7.2B]

[7.4B]

13. 3y 5z 3y 5z [7.4A] 2

16.

1 , 7 4

22. 6x 1 6x 5 4

25. 3 x 6 2 [7.4D]

2

17. 4x x 6 x 1 [7.2B]

[7.5A]

19. a 12 2a 5 [7.3B]

4

10. n2n 3 n 1 [7.2B]

3

12. 2b2b 7 3b 4 [7.4D]

18. 4a 3b 16a 2 12ab 9b 2

width is 60 yd. [7.5B]

3. Nonfactorable over the integers [7.3A]

5. 7y32y6 7y3 1 [7.1A]

[7.4C]

20. 7, 3 [7.5A]

23. 2x 5 5x 2y [7.1B]

[7.4C]

26. x 5 3x 2 [7.3B]

28. The distance is 20 ft. [7.5B]

27. The length is 100 yd. The

29. The width of the frame is 1.5 in. [7.5B]

30. The side of the original garden plot was 20 ft. [7.5B]

1. b 6 a 3 [7.1B]

2. 2y 2 y 8 y 1

5. a 16 a 3 [7.2A] 9. 5 x2 9x 3 [7.1A]

6. 2x3x2 4x 5 [7.1A] 10. p 6 2 [7.4A]

13. b 4 b 4 [7.4A]

11. 3, 5 [7.5A]

17. p 1 x 1 [7.1B]

the integers [7.3A]

20. x 12 x 3 [7.2A]

4. 2x 1 3x 8 [7.3A] 1 1 2 2

7. x 5 x 3 [7.2A]

14. 3y 2x 1 x 1 [7.3B]

[7.4C]

[7.5A]

3. 4 x 4 2x 3 [7.3B]

2

16. 2a 2 5 3a 2 1 3 23. , 7 2

[7.2B]

8. ,

12. 3 x 2y 2 [7.4D]

15. 3x 2 9x 2 6x 4 18. 3a 5 a 5 [7.4D]

21. 2a 3b

2

24. The two numbers are 7 and 3. [7.5B]

[7.5A]

[7.4A]

[7.4B] 19. Nonfactorable over

22. 2x 7y 2x 7y [7.4A]

25. The length is 15 cm. The width is 6 cm. [7.5B]

CUMULATIVE REVIEW EXERCISES 1. 7 [1.1C]

2. 4 [1.3A]

3. 7 [1.4A]

4. 15x2 [1.4C]

5. 12 [1.4D]

6.

2 3

[2.1C]

7.

7 4

[2.2B]


CHAPTER 7 TEST


8. 3 [2.2C]

9. 45 [2.1D]

10. 1 [4.2A]

y

11.

y

12.

4

4

2 –4

–2

2

0

2

4

x

–4

–2

–2

26. The width is 15 ft. [3.2A] at 4%. [5.1C]

y6 x8

and 7 [7.5A]

25. The third angle measures 59°. [3.1C]

27. The pieces are 4 ft long and 6 ft long. [2.3B]

29. The distance to the resort is 168 mi. [2.4C]

[4.3B]

21. 5xy 2 (3 4y2) [7.1A]

5 and 4 [7.5A] 2

24.

x

4

17. x3 3x2 6x 8 [6.3B]

16. 9a6b4 [6.1A]

20. (a b)(3 x) [7.1B]

[6.1B]

2

–4

[4.3A]

15. (1, 2) [5.2A]

[5.1B]

0 –2

–4

2 13. y x 6 [4.5A] 14. (1, 6) 3 21 18. 4x 8 [6.4B] 19. 2x 3 2 22. (x 7y)(x 2y) [7.2A] 23. 3

A23

28. $8000 is invested

30. The length of the base of the triangle

is 12 in. [7.5B]

Answers to Chapter 8 Selected Exercises PREP TEST 1 6

1 24

5 24

1 3

2.

10 8. 7

9. The rate of the first plane is 110 mph, and the rate of the second plane is 130 mph. [2.4C]

[2.2C]

3.

3 2

1. 50 [1.2C]

[1.2D]

[1.2D]

4.

[1.2C]

5.

[1.2C]

6.

7. 2 [2.2C]

[1.4A]

SECTION 8.1 3.

3 4x

23.

5. x2 x5

39.

x2( x 1) y( x 3)

55.

x5 x 12

73. 1

1 x3

25.

7. 1 2( x 2) x3

41. 59.

75.

9.

x8 x4

61.

11.

2x 1 2x 3

3 4x

2 x

17.

y2 y3

19.

x5 x4

35ab2 24x2y

33.

4x3y3 3a2

35.

3 4

n 10 n7

51.

x( x 2) 2( x 1)

15.

x7 x6

31.

43. ab2

45.

x5 x4

47. 1

3a 2

65.

x2( x 4) y2( x 2)

4 3

63.

2n 1 2n 3

81.

x x8

2x2 4x 3x 6 , 6x2 x 2 6x2 x 2

5.

77.

a b

13.

29.

27.

y( x 1) x2( x 10)

7a3y2 40bx

2 3y

83.

67.

49. x( x 2) y( x 6)

69.

3by ax

21.

x4 x3

37. ab2

71.

53.

x2 x6

( x 6) ( x 3) ( x 7) x 6)

n2 n3


SECTION 8.2 1.

9y3 17x , 12x2y4 12x2y4

9.

2x x2 4x 3 , x 3 x 3 x 3 x 3

3.

11.

3x 1 6x3 30x2 , 2x x 5 2x x 5

6 5x 10 y , 6 x 2y x 2y 6 x 2y x 2y

15.

x3 2x 4 , x 2 x2 2x 4 x 2 x2 2x 4

19.

12x2 8x 6x2 9x , 2x 3 2x 5 3x 2 2x 3 2x 5 3x 2

17.

23.

2x2 10x 2x 6 x1 , , x 5 x 3 x 5 x 3 x 5 x 3

31.

12ab 9b 8a 30a2b2

33.

5 16b 12a 40ab

35.

7 12x

7.

6x2 9x 10x2 15x , 2x 3 2x 3 2x 3 2x 3

13.

3x2 3x 5x2 5x , x 1 x 1 2 x 1 x 1 2

2x2 6x x2 x 2, x 1 x 3 x 1 x 3 2

21. 25.

3x2 x 4 5 4x2 6x , , 3x 4 2x 3 3x 4 2x 3 3x 4 2x 3

x5 2xn1 2x , n n x 1 x 2 x 1 xn 2

37.

n

2xy 8x 3y 10x2y2

39.

27.

a2a 13 a 1 a 2

1 2x2

29. 41.

1 x2

5x2 6x 10 5x 2 2x 5

A24

Chapter 8

43.

a ba b

55.

3xn 2 xn 1 xn 1

a2 18a 9 aa 3

45.

57. 1

x1 2x 1

67. 1

69.

83. a.

b6 3b 2

71.

x2 52x 160 4 x 3 x 3

59.

1 2x 1

73.

6x 1 x2x 1

b.

17x2 20x 25 x6x 5

47.

1 x 4

6 x 3 2 x 3

3x 1 4x 1

61. 75.

2

85. a. f 4

49.

3a 3a

63.

51.

2 x 1 x 2 2

25x 3 x 4 x 3 x 3

2x2 5x 2 x 2 x 1

77.

53.

79.

65.

x2 x3

ba b 2a

2 2 ; g4 4; S4 4 ; Yes, f 4 g4 S4 3 3

5x2 17x 8 x 4 x 2

2 x2

81.

b. Sa f a ga

SECTION 8.3 3.

5 23

21. 35.

2 5

5. x4 2x 3

49. a.

37.

2 2x

9.

a a2

aa2 a 1 a2 1

23.

3x 1 x5

x x1

7.

2a 2 7a 4

11. x1 x4

25. xy xy

39.

a1 a1

27.

41.

13.

2 5

15.

x 1 x 1 x2 1

2x x2 1

43.

1 2

17.

29.

a2 1 2a

x2 x 1 x2 x 1

1 x x h

45.

x2 x3

31.

3x 2 x2

19.

47.

x2 x1 x3 x4

33.

3n2 12n 8 nn 2 n 4

x3 x2

b. 1

c.

7. 9

9. 1

SECTION 8.4 3. 3

5. 1

27. No solution

11.

1 4

13. 1

3 2

31. , 4

29. 4, 2

15. 3

17.

1 2

19. 8 39.

33. 3

35. 4

37. 0

13. 6

15. 4

17.

23. 1

21. 5 2 5

41. 0,

25. 5

2 3

SECTION 8.5 3. 9

5. 12

amendment.

7. 7

9. 6

21. The distance between the two cities is 175 mi.

25. The person is 67.5 in. tall. 6.7 cm.

11. 1

19. 20,000 voters voted in favor of the

23. The sales tax will be $97.50 higher.

27. There are approximately 75 elk in the preserve.

31. The height is 2.9 m.

6.25 cm.

2 3

39. The length of DA is 6 in.

45. The first person won $1.25 million.

29. The length of side AC is

35. The area is 48 m2.

33. The perimeter is 22.5 ft.

41. The length of OP is 13 cm.

37. The length of BC is

43. The distance across the river is 35 m.

47. The player made 210 foul shots.

SECTION 8.6

13. y 3x 8 25. t

d r

39. w

3. y 4x 3 2 3

15. y x 3 PV nR

29. l

41. a. S

F BV B

27. T

A S1

3 2

5. y x 3

7. y

17. x 6y 10 P 2w 2

31. b1

2 x2 5

19. x

2A hb2 h

2 7

9. y x 2

1 y3 2

33. h

b. The required selling price is $180.

1 3

11. y x 2

3 4

21. x y 3 3V A

35. S C Rt

23. x 4y 3 37. P

A 1 rt

c. The required selling price is $75.

SECTION 8.7 3. It will take 2 h to fill the fountain with both sprinklers working. take 3 h to remove the earth.

7. With both computers working, it would take 30 h to solve the problem.

take 30 min to cool the room with both air conditioners working. the tank.

5. With both skiploaders working together, it would

13. It would take the apprentice 15 h to construct the wall.

complete the wiring.

was 360 mph.

15. It will take the second technician 3 h to

17. It would have taken one of the welders 40 h to complete the welds.

taken one machine 28 h to fill the boxes. area was 20 mph.

21. The jogger ran 16 mi in 2 h.

19. It would have

23. The rate of travel in the congested

25. The rate of the jogger was 8 mph. The rate of the cyclist was 20 mph. 29. Camille’s walking rate is 4 mph.

9. It would

11. It would take the second pipeline 90 min to fill

31. The rate of the car is 48 mph.

27. The rate of the jet 33. The rate of the wind is


1. y 3x 10


20 mph.

35. The rate of the gulf current is 6 mph.

A25

37. The rate of the trucker for the first 330 mi was 55 mph.

39. The bus usually travels 60 mph.

SECTION 8.8 3. The pressure is 6.75 lbin2.

1. The profit is $80,000. will roll 54 ft.

5. In 10 s, the object will fall 1600 ft.

9. When the width is 4 ft, the length is 10 ft.

13. The current is 7.5 amps. 19. Inversely

7. In 3 s, the ball

11. The gear that has 36 teeth will make 30 rpm.

15. The intensity is 48 foot-candles when the distance is 5 ft.

17. y is doubled.

21. Inversely

CHAPTER 8 REVIEW EXERCISES 1.

x2 3x 10

[8.3A]

4.

by3 6ax2

[8.1B]

9.

x x7

[8.3A]

3x2 x 24x3 4x2 , (2x 3)(6x 1)(3x 1) (2x 3)(6x 1)(3x 1)

[8.2A]

11. a

T 2bc 2b 2c

b 3y 10ax

[8.1C]

6. 62 [8.5A] 10.

13. c 17.

100m i

1 x3

2. 7.

[8.2B]

(3y 2)2 ( y 1)( y 2)

[8.6A]

3.

[8.1C]

8.

22.

x6 x3

14. The equation has no solution. [8.4A]

3x 1 x5

ABC is 24 in. [8.5C]

[8.2B]

23. 10 [8.5A]

15.

[8.1A]

1 x2

[8.6A]

[8.1C]

24. 12 [8.5A]

28. The rate of the wind is 20 mph. [8.7B]

[8.1A]

12. 2 [8.4A] 2y 3 5y 7

16.

[8.2B]

8x 5 3x 4

20.

[8.1B]

25. The perimeter of triangle

26. It would take 6 h to fill the pool using both hoses. [8.7A]

is 45 mph. [8.7B]

2x4 3y7

5.

4 19. y x 2 [8.6A] 9

18. (5x 3)(2x 1)(4x 1) [8.2A]

[8.2B]

21. 5 [8.4A]

7x 22 60x

27. The rate of the car

29. The pitcher’s ERA is 1.35. [8.5B]

30. The current is 2 amps. [8.8A]

CHAPTER 8 TEST 1.

2x 3 3y 3

[8.1A]

2.

x5 x1

[8.1A]

6. 3(2x 1)(x 1) [8.2A] 9.

5 (2x 1)(3x 1)

7.

[8.2B]

no solution. [8.4A]

10.

wallpaper are needed. [8.5B]

[8.1B]

4.

3(x 2) x2 ; x(x 2)(x 2) x(x 2)(x 2) x2 4x 5 (x 2)(x 3)

14. 1 [8.5A]

cyclist is 10 mph. [8.7B]

x1 x3(x 2)

3.

[8.2B]

11.

(x 5)(2x 1) (x 3)(2x 5) [8.2A]

x3 x2

[8.1B]

2 x5

8.

[8.3A]

x5 x4

[8.1C]

[8.2B]

12. 2 [8.4A]

16. t

15. The area is 64.8 m2. [8.5C]

5.

ds r

13. The equation has

[8.6A]

17. 14 rolls of

18. It would take 10 min with both landscapers working. [8.7A]

19. The rate of the

20. The resistance is 0.4 ohm. [8.8A]



31 30

[1.3A]

7. 10 [2.1D]

2. 21 [1.4A]

3. 5x 2y

8. {xx 8} [2.5A]

[1.4B]

4. 8x 26 [1.4D]

9. The volume is 200 ft3. [3.3A]

5.

9 2

6. 12 [2.2C]

[2.2A]

10.

y 4 2 –4

–2

0

2

4

x

–2 –4

11.

3 7

[4.2A]

16. 3.5 108

12. y

3 x 2 [4.6A] 2

[6.1C]

20. ( y 6)( y 1) [7.2A]

[4.3B] 5

13. 2, 1, 1 [5.2B]

17. 4a4 6a3 2a2

[6.3A]

a b6

14. a3b7

[6.1A]

15.

18. a2 ab 12b2

[6.3B]

19. x2 2x 4 [6.4B]

21. (4x 1)(3x 1) [7.3A/7.3B]

[6.1B]

22. a(2a 3)(a 5) [7.3A/7.3B]

A26

Chapter 9

23. 4(b 5)(b 5) [7.4D] 28.

3 (2x 1)(x 1)

5 2

24. 3 and

[8.2B]

29.

32. It would cost $160. [8.5B]

x3 x5

[7.5A]

[8.3A]

2x3 3y5

25.

26.

[8.1A]

30. 4 [8.4A]

x2 x5

[8.1A]

27. 1 [8.1C]

31. The alloy contains 70% silver. [2.4B]

33. It would take 6 min to fill the tank. [8.7A]

Answers to Chapter 9 Selected Exercises PREP TEST 1. 16 [1.1D]

2. 32 [1.2E]

7. 9x2 12x 4 [6.3C]

3. 9 [1.2D]

1 12

4.

5. 5x 1 [1.4D]

[1.2C]

8. 12x2 14x 10 [6.3B]

9. 36x2 1 [6.3C]

xy5 4

6.

[6.1B]

10. 1 and 15 [7.5A]

SECTION 9.1 1. 2

3. 27

19. a7/12

1 a

21.

41. x4y y x3

4

d. False; a b n

c. True

51.

33. x3/10 53.

x

69. x4n

131. x y

133. 3xy

1/2

1/2

b

5

1 x1/2

39. y1/9

57. a5b13 75. x 2n

2

5

89. a4b2

119. x5y

4 2

37.

1 b7/8

73. y3n/2 3

17. x1/12

35. a3 55.

105. 2x2 1/3

103. b3/5

117. xy3

1 y5/2

71. x3n/2

3

e. False; a 2a b

n 1/n

1 11/12

15. y1/2

13. x

31. a

87. 2x2

101. x4/3

129. x y

1 x4

x2 y8

85. 32t5

2 3

343 125

11.

29.

49.

115. x4

113. x8 3 4

125. 3x

x3/2 y1/4

1 x

67. a a2

99. x1/3

127. 4x y

3

27.

47.

83. a3

97. 141/2

111. a2 2 1/2

x y2

65. y2 y

81. 3

1 3 x2

9. Not a real number 25. y3/2

45.

16b a1/3

63.

1 y

23.

2

79. x4ny2n 95.

7. 4

43. x6y3z9

17/2

61.

1 9

5.

77. x2nyn 4

93. 4x 3 3

91. a6b12 107. 3x5 1/2

m2 4n3/2

59.

109. 3xy2/3

121. 4a2b6

123. Not a real number

135. 2ab2

137. a. False; 2

b. True

n/m

f. False; a

SECTION 9.2 3. 2ab42a

2 3

5. 3xyz25yz

4

13. abc ab2

3

25. 6ab 3ab 3ab3ab 3

65. 672x2y2

67. 4a3b3a

23y 3y

99. 109.

3

85.

33a a

12 42 7

is in simplest form.

2y 2

89.

10 57 3

111.

3 53 3

103.

3 7y 2y 1 4y 2

73. 12x y 3 2 59y 3y

91.

7x 21 x9

93.

47. xy x

6

3

e. False; 2 3 is in simplest form.

77. y5y

b2a 2a2

95.

63. 84 165

79. b13b

15x 5x

81.

2 2

97. 2 22 107.

c. False; x2/3

b. True

f. True

49. 8xyx

61. x 6x 9

105. 6 3 22 23

113. a. False; 432

35. 5bb

3

45. 24

59. 4x 8x

71. x y2

23. 2xy2y

3

43. 16

57. x 2x

69. 85 87.

101.

8a 10ab 3b 16a 9b

41. 4ab2b

55. 6

3

11. 5yx2y

33. 172 155

31. 3a2a

4

39. 2y2x 4

53. 2ab3a2b

21. 32b 53b 4

29. 8b2b2

3

51. 2x2y2

19. 22 3

27. 2

37. 8xy2x 2xyxy

83.

17. 6x

15. 2x2yxy

2

3

9. a5b2ab2

7. Not a real number

17 55 4

d. False; x y

4

115. a b

SECTION 9.3 1. 2

3. 9

5. 7

7.

13 3

moon, an object will fall 24.75 ft in 3 s. pendulum is approximately 7.30 ft.

11. 7

9. 35

13. 12

15. 9

17. 2

19. 2

25. The HDTV screen is approximately 7.15 in. wider. 29. 2

31. r

3

3V 4

21. 1

23. On the

27. The length of the


1. x2yz2yz

A27


SECTION 9.4 5. 7i2

3. 2i

19. 6 6i

21. 19 7i2

35. 17 i 53.

9. 4 2i

7. 3i3

23. 62 3i2

37. 8 27i

25 5 i 5 5

55.

11. 23 3i2

39. 1 3 11 i 10 10

6 7 i 5 5

57.

1 3 45. i 4 2

43. 3i

41. 1

29. 32

27. 4

25. 63

15. 8 i

13. 410 7i3

59. a. Yes

17. 8 4i

31. 4 12i

2 10 47. i 13 13

33. 2 4i

4 2 49. i 5 5

51. i

b. Yes

CHAPTER 9 REVIEW EXERCISES 1. 20x2y2 [9.1A]

1 7. 5 x

6. 2 [9.3A] 11.

3. 39 2i [9.4C]

2. 7 [9.3A] [9.1A]

xx x2 2x 22 x2 3 3 3

[9.2D]

2 5 i 3 3

12.

20. 2 7i [9.4D]

21. 62 [9.4C]

25. 12 10i [9.4B] 30. 7x3y8 [9.1C]

14. 42 8i2 [9.4B]

4

5

19. 2ab22a3b2 [9.2A]

18. 3x3 [9.1B]

23. 3a2b3 [9.1C]

22. 30 [9.3A]

26. 31 106 [9.2C]

10. 2a2b2b [9.2B]

13. 3ab32a [9.2A]

[9.4D]

17. 7 3i [9.4C]

16. 4xy x2 [9.2C]

5. 63 13 [9.2C]

9. 2a2b4 [9.1C]

[9.2D]

2 3

15. 5x y 2x2y [9.2B]

242 ft. [9.3B]

83y 8. 3y

4. 7x2/3 y [9.1B]

27. 6x23y [9.2B]

28.

24. 5i2 [9.4A]

1 3

[9.1A]

31. The amount of power is approximately 120 watts. [9.3B]

29.

1 a10

[9.1A]

32. The distance required is

33. The distance is approximately 6.63 ft. [9.3B]

CHAPTER 9 TEST 1.

1 3/4 x 2

6. r1/6

3

[9.1A]

13. 10 2i

[9.2C]

[9.1A]

3. 3y2

8. 2xy2

7. 4 [9.3A]

12. 14 103 64x3 17. y6

5

2. 2x2y2x [9.2B]

[9.1B]

9. 4x3

[9.1C]

[9.4C]

[9.2D]

4 7 19. i 5 5

4x2 y

[9.2D]

24. 4 [9.4C]

3

23.

15. 8a2a 20. 3

[9.4D]

5. 4x 4xy y

[9.4C]

10. 3 2i

[9.2C]

14. 2 [9.2D]

x xy 18. xy

22. 3abc2ac [9.2A]

4. 18 16i

[9.1B]

11. 4x2y32y

[9.4B]

16. 2a ab 15b

[9.2B]

b3 21. 8a6

[9.3A]

[9.2C] [9.2A] [9.2C]

[9.1A]

25. The distance is 576 ft. [9.3B]

CUMULATIVE REVIEW EXERCISES 1. 92 [1.3A] 7.

1 7 , 3 3

8. {x6 x 3} [2.6B]

[2.6A]

m

y

11. 4


3. 10x 1 [1.4D]

2. 56 [1.4A]

3 2

0

2

4

2 3

6. {xx 1} [2.5A]

[2.2C]

13. y

10. The volume is 14 ft3. [3.3A] 1 7 x 3 3

[4.5B]

2 –4

–2

0

2

4

x

–2

[4.3B]

3, 2 [5.2A]

23.

5.

4

x

–4

18. C R nP

[2.2A]

y

12.

–2

14.

3 2

9. The area is 187.5 cm2. [3.2B]

b3 –4

4.

15.

[8.6A]

1 3 i [9.4D] 5 5

invested at 3.5%. [5.1C]

2x 2 y 2 19.

[6.1B] y5 x4

[9.1A]

24. 20 [9.3A]

–4

[4.7A]

16. (9x y)(9x y) [7.4A] 20. x10x [9.2B]

17. x(x2 3)(x 1)(x 1) [7.4D]

21. 13 73 [9.2C]

25. The length of side DE is 27 m. [8.5C]

27. The rate of the plane was 250 mph. [8.7B]

Earth from the moon. [6.1D] The interest rate is 8%. [4.4A]

22. 6 2 [9.2D]

26. $2500 is

28. It takes 1.25 s for light to travel to

29. The slope is 0.08. The slope represents the simple interest rate on the investment. 30. The periscope must be approximately 32.7 ft above the water. [9.3B]

A28

Chapter 10

Answers to Chapter 10 Selected Exercises PREP TEST 1. 32 [1.2F]

2. 3i [9.4A]

7. 3x 2 3x 2 [7.4A]

3.

2x 1 x1

[8.2B]

4. 8 [1.4A]

5. Yes [7.4A]

6. 2x 1 2 [7.4A]

[1.5C]

9. 3 and 5 [7.5A]

10. 4 [8.5A]

8. −5 −4 −3 −2 −1

0

1

2

3

4

5

SECTION 10.1 5. 4x2 5x 6 0; a 4, b 5, c 6

3. 2x2 4x 5 0; a 2, b 4, c 5 11. 2 and 3 2 3

27. and

13. 3

9 2

29. 4 and

41. c and 7c

1 4

43. b and

53. x2 7x 10 0

17. 2 and 5

15. 0 and 2

31. 2 and 9 b 2

45.

65. x2 3x 0

73. 9x 4 0

75. 6x 5x 1 0

2

2a and 4a 3

85. 2i and 2i

95. 53 and 53

115. x2 1 0

35. 5 and 2

a and 3a 3

49.

57. x2 5x 6 0 77. 10x 7x 6 0

87. 2 and 2

89.

119. x2 2 0

59. x2 9 0

51.

91. 7i and 7i

39. 2b and 7b

4a a and 3 2

81. 50x2 25x 3 0

93. 43 and 43

103. 7 and 3

3b 3b and a a

1 3

71. 3x2 11x 10 0

2 92 2 92 111. and 3 3

121.

3 2

25. 4 and

61. x2 8x 16 0

79. 8x 6x 1 0

101. 6 and 0

109. 3 3i5 and 3 3i5

117. x2 8 0

3y y and 2 2

2

9 9 and 2 2

99. 5 and 7

1 23. and 2 4

37. 4 and

69. 4x2 5x 6 0

2

97. 3i2 and 3i2

107. 5 6 and 5 6

47.

67. 2x2 7x 3 0

2

83. 7 and 7

3 4

33. 2 and

55. x2 6x 8 0

63. x2 5x 0

19. 2 and 5

9. 5 and 5

7. 0 and 4

3 21. and 6 2

105. 0 and 1 113. x2 2 0

123. a 2 and a 2

125.

1 2

SECTION 10.2 1. 1 and 5

3. 9 and 1

13. 3 and 8 23. 3 and 5

25. 2 3i and 2 3i

41.

3 and 1 2

51. 0.293 and 1.707

9. 3 2 and 3 2

1 5 1 5 19. and 2 2

27. 3 2i and 3 2i

33. 1 2i3 and 1 2i3

2 14 2 14 and 2 2

49. 3.236 and 1.236

7. 2 11 and 2 11

3 5 3 5 17. and 2 2

15. 9 and 4

1 17 1 17 31. and 2 2

39.

5. 3

11. 1 i and 1 i

21. 3 13 and 3 13

29. 1 32 and 1 32

1 1 35. i and i 2 2

43. 1 5 and 1 5

45.

37.

1 1 1 1 i and i 3 3 3 3

1 and 5 2

55. a and 2a

53. 0.809 and 0.309

47. 2 5 and 2 5 57. 5a and 2a

SECTION 10.3 3. 2 and 5 13.

1 3 and 4 2

5. 9 and 4

15. 7 35 and 7 35

21. 1 2i and 1 2i 29.

7. 4 222 and 4 222

23. 2 3i and 2 3i

3 3 33 33 i and i 4 4 4 4

17.

1 3 1 3 and 2 2

25.

31. 0.394 and 7.606

37. Two complex number solutions

9. 8 and 3

11.

19. 1 i and 1 i

1 11 1 11 and 2 2

33. 4.236 and 0.236

39. One real number solution

5 33 5 33 and 4 4

27.

35. 1.351 and 1.851

41. Two real number solutions

43. No. The arrow does not reach a height of 275 ft. (The discriminant is less than zero.)

47. p p 1

49. 2i and i

3 1 3 1 i and i 2 2 2 2

45. p p 9


59. No; the ball will have gone only 197.2 ft when it hits the ground.


A29

SECTION 10.4 3. 2, 2, 2, and 2

1. 2, 2, 3, and 3 13. 16

15. 1 and 512

29. 2

31. 1

2 2 3 3

17. , , 1, and 1

45.

1 1 and 3 2

47.

2 and 6 3

23. 1 and 2

21. 9

11. 4i, 4i, 2, and 2

4 and 3 3

51.

27.

25. 0 and 2

1 1 7 7 i and i 2 2 2 2

39.

49.

9. 2i, 2i, 1, and 1

7. 16

19. 3

37. 1 and 10

35. 3

33. 1

43. 1 and 0

5. 1 and 4

1 and 3 4

1 and 2 2

41. 3 and 1 55. 5 or 5

53. 9 and 36

SECTION 10.5 3. 7.

x x 2 or x 4

−5 −4 −3 −2 −1 0 1 2 3 4 5

x 3 x 4

−5 − 4 −3 −2 −1 0 1 2 3 4 5

11.

x 1 x 3

−5 −4 −3 −2 −1 0 1 2 3 4 5

19. x x 4 or x 4

−4

−2

29.

37. 0

2

23.

−4

4

13.

x x 2 or 1 x 3

−5 − 4 −3 −2 −1 0 1 2 3 4 5

x

x x 2 or x 4

−5 −4 −3 −2 −1 0 1 2 3 4 5

1 3 x 2 2

25.

1 x 1 2

x

x x 2 or 1 x 3

−5 −4 −3 −2 −1 0 1 2 3 4 5

17.

21. x 3 x 12

27. x x 1 or 1 x 2 35.

9.

x x 1 or x 2

−5 −4 −3 −2 −1 0 1 2 3 4 5

x 4 x 1 or x 2

−5 − 4 −3 −2 −1 0 1 2 3 4 5

15.

5.

31. x 2 x 3

x x 1 or x

5 2

33. x x 5 or 4 x 1 39.

−2

0

2

−4

4

−2

0

2

4

SECTION 10.6 1. The height is 3 cm. The base is 14 cm. 5. The maximum speed is 33 mph.

3. The dimensions of Colorado are approximately 272 mi by 383 mi.

7. The rocket takes 12.5 s to return to Earth.

approximately 72.5 kmh and still be able to stop within 150 m. It would take the smaller pipe 12 min to fill the tank. 15. The rowing rate of the guide is 6 mph.

9. A driver can be going

11. It would take the larger pipe 6 min to fill the tank.

13. The rate of the wind was approximately 108 mph.

17. The radius of the cone is 1.5 in.

CHAPTER 10 REVIEW EXERCISES 1. 0 and

3 2

2. 2c and

[10.1A]

5. 3 and 1 [10.2A]


12. 2i and 2i [10.2A] 15.

18.

5 4

x 3 x [10.4B]

5 2

[10.1A]

13.

[10.2A]

11 73 11 73 and 6 6

[10.5A]

16.

x x 4 or

19. 1 and 3 [10.4C]

3 4 and 4 3

[10.3A]

3 x 2 2

20. 1 [10.4C]

[10.4C]

26.

[10.3A]

[10.5A]

8.

[10.1C]

1 1 31 31 i and i 2 2 2 2

[10.2A]

17. 64 and 27 [10.4A]

x x 2 or x 2

3

21. −5 −4 −3 −2 −1 0 1 2 3 4 5

[10.5A]

23. 4 [10.4B] [10.4C]

24. 5 [10.4B]

27. The width of the rectangle

28. The integers are 2, 4, and 6 or 6, 4, and 2. [10.6A]

29. Working alone, the new computer can print the payroll in 12 min. [10.6A] in calm water is 6 mph. [10.6A]

1 1 2i and 2i 2 2

11. 1 i7 and 1 i7

11 129 11 129 and 2 2

is 5 cm. The length of the rectangle is 12 cm. [10.6A]

4.

14. Two real number solutions [10.3A]

x x 3 or 2 x 4

1

22.

3 249 3 249 and 10 10

7.

10. 12x2 x 6 0 [10.1B]

−5 −4 −3 −2 −1 0 1 2 3 4 5

25.

3. 43 and 43 [10.1C]

7 27 7 27 and 7 7

9. x2 3x 0 [10.1B]

[10.3A]

[10.5A]

6.

c 2

30. The sculling crew’s rate of rowing

A30

Chapter 11

CHAPTER 10 TEST 1. 4 and

2 3

2.

[10.1A]

5. 2 22 and 2 22 8.

1 3 1 3 and 2 2

2 3 and 3 2

3. x2 9 0 [10.1B]

[10.1A]

6. 3 11 and 3 11

[10.1C]

9. 2 2i2 and 2 2i2

[10.3A]

11. Two complex number solutions [10.3A] 15. No solution [10.4B]

1 4

12.

16. 9 and 2 [10.4C]

is released. [10.6A]

[10.3A]

[10.5A]

[10.1B]

3 15 3 15 and 3 3

[10.2A]

10. Two real number solutions [10.3A]

13. 3, 3, 1, and 1

[10.4A]

14. 4 [10.4B]

x x 4 or 2 x 4

−5 −4 −3 −2 −1 0 1 2 3 4 5

3

−5 − 4 −3 −2 −1 0 1 2 3 4 5

[10.4A]

7.

17.

x 4 x 2

18.

[10.2A]

4. 2x2 7x 4 0

[10.5A]

19. The ball hits the basket approximately 1.88 s after it

20. The rate of the canoe in calm water is 4 mph. [10.6A]

CUMULATIVE REVIEW EXERCISES

2. x2 x

1. 14 [1.4A] 6.

5 , 0 , 0, 3 2

10 3

3. The volume is 54 m3. [3.3A]

[2.6B]

7. y x 1 [4.6A]

[4.3C]

8. (1, 1, 2) [5.2B]

4.

9.

7 3

5.

[4.2A]

3 2

{4.4A]

y 4 2 –4 –2 0 –2

2

4

x

–4

[5.4A] 11. x2 3x 4

10. The height of triangle DEF is 16 cm. [8.5C] 13. (2x 5)(3x 4) [7.3A/7.3B]

is 9

x 2

18. 8 14i [9.4C]

3 and 1 [8.4A] 2

19. 0, 1 [9.3A]

25 23 in. The upper limit is 9 in. [2.6C] 64 64

23. The slope is

15.

[8.1B]

[6.4B]

16. b

20. 2, 2, 2, 2

2S an n

[10.4A]

[8.6A] 21. The lower limit

22. The area is (x2 6x 16) square feet. [6.3D]

25,000 . The building decreases $8333.33 in value each year. [4.4A] 3

up on the house. [9.3B]

12. 3xy(x2 2xy 3y2) [7.1A]

24. The ladder will reach 15 feet

25. There are two complex number solutions. [10.3A]

Answers to Chapter 11 Selected Exercises PREP TEST 1. 1 [1.4A]

2. 7 [1.4A]

6. 2 3 and 2 3 [10.3A] The relation is a function. [4.2A]

4. h2 4h 1 [4.2A]

3. 30 [4.2A] 7. y

1 x 2 [8.6A] 2

9.

[4.3B]

y

2 –2

0 –2 –4

2

4

2 and 3 [10.1A] 3

8. Domain: 2, 3, 4, 6; Range: 4, 5, 6.

4

–4

5.

x


17. 1 a [9.1A]

14.

6 3x 4

A31


SECTION 11.1 5. 5

7. x 7

9.

11.

y

–4

y 8

4

4 –8

13.

y

8

4

2

0

4

8

x

–4

–2

0

2

4

x

–8

–4

0

–4

–2

–4

–8

–4

–8

Vertex: 1, 5

Vertex: 1, 2

Axis of symmetry: x 1


Vertex:

4

x

8

25 1 , 2 4

Axis of symmetry: x 15.

17.

y

–4

–2

19.

y

21.

y

y

4

8

4

4

2

4

2

2

0

2

4

x

–8

–4

0

4

8

x

–4

–2

0

2

4

x

–4

–2

0

–2

–4

–2

–2

–4

–8

–4

–4

Vertex:

3 1 , 2 4

Vertex:

Axis of symmetry: x 23.

3 2

3 9 , 2 2

Axis of symmetry: x

3 2

25. Domain: x x real numbers Range: yy 5

y 4

1 2

2

4

x

Vertex: 0, 1

Vertex: 2, 1



27. Domain: x x real numbers Range: y y 0

2 –4

–2

0

2

4

x

–2 –4

Vertex: 1,

5 2

Axis of symmetry: x 1 29. Domain: x x real numbers

33. 3, 0 and 3, 0

35. 0, 0 and 2, 0

37. 4, 0 and 2, 0

Range: y y 7 39.

1 , 0 and 3, 0 2

3 41 2 Copyright © Houghton Mifflin Company. All rights reserved.

57. Two

49. 0 and 59. One

41. 2 7, 0 and 2 7, 0 4 3

51. i2 and i2 61. No x-intercepts

73. a. Minimum

b. Maximum

79. Maximum value:

9 8

will give the maximum revenue. is 24.36 ft.

65. No x-intercepts

75. Minimum value: 2

11 4

83. Maximum value:

9 4

45. 3 55.

47.

67. Two

69. 4, 0 and 5, 0

77. Maximum value: 3 85. Minimum value:

93. The diver reaches a height of 13.1 m above the water.

101. The two numbers are 10 and 10.

3 41 and 2

1 41 1 41 and 4 4

89. The maximum value of the function is 5.

2 97. The water will land at a height of 31 ft. 3

sign 44 ft away. 105. k 16

63. Two

c. Minimum

87. Parabola b has the greatest maximum value.

1 1 47 47 i and i 6 6 6 6

53.

81. Minimum value:

43. No x-intercepts

1 12

91. A price of $250

95. The minimum height

99. The car can be traveling 20 mph and still stop at a stop 103. The length is 100 ft and the width is 50 ft.

A32

Chapter 11

SECTION 11.2 1. Yes

3. No

5. Yes Domain: x x real numbers Range: y y 0

y

7. 4


y

9. 4 2

2 –4 –2 0 –2

2

x

4

–4 –2 0 –2

2

x

4

–4

–4

Domain: x x 4 Range: y y 0

y

11. 4

4 2

2 –4 –2 0 –2

2

x

4

–4 –2 0 –2

2

x

4

–4

–4

Domain: xx 2 Range: y y 0

y

15.


y

13.

4

Domain: x x real numbers Range: y y 0

y

17. 4 2

2 –4 –2 0 –2

2

x

4

–4 –2 0 –2

2

x

4

–4

–4

19. a 18

21. 17

23. f14 8

25. x 2 x 2

27. x 1

SECTION 11.3 1. 5

3. 1

25. 13 43. 5

5. 0 27. 29

7.

9.

2 3

29. 8x 13

11. 2 31. 4

47. 3x 3x 5

45. 11

sells each camera for $83.20. c. rd p (The cost is less.)

b. $83.20

15. 8

13. 39

51. 27

37. 1

4 5

19. 2

21. 7

41. x2 1

39. 5

53. x 6x 12x 8 3

2

c. When 5000 digital cameras are manufactured, the camera store

57. a. dr p 0.90p 1350 61. 2

59. 0

17.

35. x 4

33. 3

49. 3

2

16000 55. a. SMx 80 x

73. 6

29 4

b. rd p 0.90p 1500 67. 2 h

65. h2 6h

63. 7

69. 2a h

71. 1

75. 6x 13

SECTION 11.4 5. No

7. Yes

9. No

11. No

21. 2, 0 , 5, 1 , 3, 3 , 6, 4

23. No inverse

29. f 1 x 2x 2

1 2

39. f 1 x 53. Yes

31. f 1 x x 1

1 2 x 5 5

55. Yes

57.

25. f 1 x

33. f 1 x

1 1 x 6 2

41. f 1 x

17. 0, 1 , 3, 2 , 8, 3 , 15, 4

13. No

43.

5 3

3 x6 2

45. 3

59.

y

1 x2 4

27. f 1 x

19. No inverse

1 x2 2

35. f 1 x 3x 3

47. Yes; Yes

49. Yes

61.

y

37. f 1 x 51. No

y (4, 5)

4

4 (2, 2)

(−4, 2)

2

(0, 1) (−2, 0)

(1, 0)

–4 –2 0 2 – 2 (0, −2) –4

65. 5

67. 0

69. 4

4

x

(−3, 1)

4 (1, 3)

–4 –2 0 2 4 (0, −2) (−1, −1) –2 –4

(5, 4)

(−1, 2) 2 (−2, 1)

2

(−2, 0)

(1, −3) (2, −4)

x

–4 –2 0 –2 –4

(3, 1)

2

4

(2, −1) (1, −2)

x

1 5 x 2 2


3. Yes


A33

CHAPTER 11 REVIEW EXERCISES 1. Yes [11.2A]

2. Yes [11.4A]

3.

4.

y

y 4

4

2

2 –4 –2 0 –2

2

x

4

–4 –2 0 –2

Domain: xx real numbers Range: y y real numbers

9.

17 4

6. 0, 0 and 3, 0 [11.1B]

[11.1C]

14.

10. 5 [11.3B]

7.

15. 4

2

2 2

x

4

Range: y y 0 [11.1B]

12. 12x2 12x 1 [11.3B]

13. Yes [11.4B] y 4 2

–4 –2 0 –2

2

4

x

–4 –2 0 –2

2

x

4

–4

–4

–4

[11.2A]

8. 3 [11.1C]

16.

y

4

–4 –2 0 –2

Domain: xx 4 [11.2A]

1 1 5 5 i and i 3 3 3 3

11. 10 [11.3B]

y

x

4

–4

–4

5. Two [11.1B]

2

Domain: xx real numbers

Vertex: 1, 2

Domain: xx real numbers

Range: y y 5

Axis of symmetry: x 1 [11.1A]

Range: y y 3 [11.2A]

17. No [11.4A] [11.3B]

[11.1A]

19. 9 [11.3A]

18. 7 [11.3A] 1 6

23. f 1 x x

2 3

20. 70 [11.3A]

24. f 1 x

[11.4B]

3 x 18 2

12 7

21.

[11.3A]

22. 6x2 3x 16

25. f 1 x 2x 16

[11.4B]

[11.4B]

26. The dimensions are 7 ft by 7 ft. [11.1D]

CHAPTER 11 TEST 1.

3 3 23 23 i i and 4 4 4 4

4. 2 [11.3A]

[11.1B]

6.

5. 234 [11.3A]

9. f gx 2x2 4x 5 [11.3B]

y


13 2

11. f 1 x

14. Yes [11.4B]

3. Two real zeros [11.1B] 8. 5 [11.3B]

1 1 x 4 2

y

18.

4

4

4

2

2

2

–4 –2 0 –2

2

x

4

–4 –2 0 –2

–4

4

x

–4 –2 0 –2

Range: y y 0

[11.2A]

2

4

x

–4

–4

Domain: x x 3 19. No [11.4A]

2

12. f 1 x 4x 16 [11.4B]

[11.4B]

15. 100 [11.1D]

y

17.

[11.1B]

7. 5 [11.3A]

[11.3A]

10. 9 [11.1C]

13. 6, 2 , 5, 3 , 4, 4 , 3, 5 [11.4B] 16.

3 41 3 41 and 2 2

2.

Domain: x x real numbers

Domain: x x real numbers

Range: y y 2

Range: y y real numbers

[11.2A]

[11.2A]

20. Dimensions of 50 cm by 50 cm would give a maximum area of 2500 cm2. [11.1D]


23 4

[1.4A]

[1.5C]

2. –5 –4 –3 –2 –1

0

5. The set of all real numbers [2.6B]

1

2

3

4

6.

5

a10 12b4

[6.1B]

3. 3 [2.2C]

4. x x 2 or x 3 [2.5B]

7. 2x3 4x2 17x 4 [6.3B]

8.

1 ,3 2

[5.2A]

A34

Chapter 12

9. xy x 3y x 2y [7.2B] 5 12. 2x 1

10. 3 and 8

13. 2 [8.4A]

[8.2B]

11. x x 3 or x 5 [10.5A]

[7.5A]

14. 3 2i [9.4D]

15.

Vertex: 0, 0 [11.1A] Axis of symmetry: x 0

y 4 2 −4 −2 0 −2

2

x

4

−4

16.

3 2

17. y 2x 2 [4.5B]

[4.7A]

y

18. y x

7 2

[4.6A]

4 2 −4 −2 0 −2

2

4

x

−4

19.

1 1 3 3 i and i 2 6 2 6

24. 2 [9.3A]

[10.3A]

1 3

26. f 1 x x 3 [11.4B]

25. 10 [11.3B]

$3.96. [2.4A]

21. 3 [11.1C]

20. 3 [10.4B]

22. 1, 2, 4, 5 [4.2A]

27. The cost per pound of the mixture is

28. 25 lb of the 80% copper alloy must be used. [2.4B]

required. [8.5B]

29. An additional 4.5 oz of insecticide are

30. It would take the larger pipe 4 min to fill the tank. [10.6A]

the string 24 in. [8.8A]

23. Yes [4.2A]

31. A force of 40 lb will stretch

32. The frequency is 80 vibrationsmin. [8.8A]

Answers to Chapter 12 Selected Exercises PREP TEST 1.

1 9

[6.1B]

3. 3 [6.1B]

2. 16 [6.1B]

7. 6326.60 [1.4A]

6. 2 and 8 [10.1A]

[11.1A]

y

8.

5. 6 [2.2B]

4. 0; 108 [4.2A/6.1B]

4 2 −4

−2

0

2

4

x

–2

SECTION 12.1 3. c

5. a. 9

b. 0.3679 17. a. 0.1353

b. 1

c. 1.2840 b. 0.1353

c.

1 9

7. a. 16

13. a. 54.5982 c. 0.0111

b. 4

c.

b. 1

1 4

9. a. 1

c. 0.1353

15. a. 16

y

19.

b.

1 8

b. 16

21. 4

2 –2

0

c. 1

y

4

–4

11. a. 7.3891

c. 16

2 2

4

x

–4

–2

0

–2

–2

–4

–4

2

4

x


–4


y

23.

y

25.

4 2 –4

–2

y

27.

4 2

0

2

x

4

–4

–2

4 2

2

0

2

x

4

–4

–2

0

2

x

4

–4

–2

0

–2

–2

–2

–2

–4

–4

–4

–4

31. b and d

33. (0, 1)

35. No x-intercept; (0, 1)

y

37.

–4

41. a.

y

29.

4

–2

2

4

2

2 2

4

x

x

y

39.

4

0

4

–4

–2

0

–2

–2

–4

–4

2

4

x

b. The point (2, 27.7) means that after 2 s, the object is falling at a speed of 27.7 fts.

v 30 20 10 0

1

2

3

4

t

5

SECTION 12.2 3. log525 2

5. log4

17. bv u

19. 4

39. 316.23

41. 0.02

57. log4x2y2 71. log4

1 2 16

21. 7

73. ln

83. 2 log3 x 6 log3 y

23. 2

61. ln

x y3

75. log2

1 log3 x 2

113. 0.6309 b. True

x y2z

63. log6

xz

115. 0.2727

51. ln

77. log8 x log8 z

3

y2

103. 0.2218

35.

67. ln

x yz

1 7

37. 1

69. log2

2 2

15. e y x

55. lnx3y4

53. log7 x3

t3v2 r2

81. log b r log b s

79. 5 log3 x

89. 2 ln x ln y ln z

3 1 95. log4 x log4 y 2 2

105. 1.3863

119. 1.9266

e. False

33. 64

x4 y2

s2r 2 t4

87. 2 log2 r 2 log2 s

117. 1.6826

d. True

31. 9

65. log4

93. 2 log8 x log8 y 2 log8 z

101. 0.8451

c. False

x y

13. 102 0.01

11. 32 9

29. 4

49. log3 x3y2

85. 3 log7 u 4 log7 v

91. log5 x 2 log5 y 4 log5 z 99. log3 t

27. 0

45. 0.61

x2z2 y

9. log a w x

25. 3

43. 7.39

59. log3

x3z y2

7. log 10 x y

107. 1.0415

121. 0.6752

97.

3 1 log7 x log7 y 2 2

109. 0.8617

123. 2.6125

111. 2.1133

125. a. False

f. True


SECTION 12.3 5.

7.

y

–4

–2

11.

y

y

4

4

4

2

2

2

2

0

2

x

4

–4

–2

0

2

4

x

–4

–2

0

2

4

x

–4

–2

0

–2

–2

–2

–2

–4

–4

–4

–4

13.

15.

y

–4

9.

y

4

–2

y

17.

y

2

19.

4

4

4

2

2

2

2

2

4

x

–4

–2

0

2

4

x

–4

–2

0

2

4

x

x

y

4

0

4

−4

−2

0

–2

–2

–2

−2

–4

–4

–4

−4

2

4

x

A35

A36

Chapter 12

21.

23. a.

y

0

2 −4

−2

b. The point (25.1, 2) means that a star that is 25.1 parsecs

M 4

4

0

2

4

5

from Earth has a distance modulus of 2.

s

10 15 20 25

−4

x

−8

−2 −4

SECTION 12.4 3.

1 3

5. 3

23. 3.5850

7. 1

9. 6

25. 1.1309

47. 1.7233

11 2

9 8

13.

15. 1.1133

31. 4 and 2

49. 0.8060

5 3

33.

51. 5.6910

17. 1.0986 1 2

35.

21. 2.8074

19. 1.3222

9 2

37.

53. a.

39. 17.5327

41. 3

43. 2

s Distance (in feet)

45. No solution

29.

4 7

11.

120 80 40 0

t

1 2 3 4 Time (in seconds)

b. It will take 2.64 s for the object to travel 100 ft.

SECTION 12.5 1. The value of the investment after 2 years is $1172.

3. The investment will be worth $15,000 in approximately 18 years.

5. a. After 3 h, the technetium level will be 21.2 mg.

b. The technetium level will reach 20 mg after 3.5 h.

7. The half-life is 2.5 years. 0.098 newtoncm2.

9. a. At 40 km above Earth’s surface, the atmospheric pressure is approximately

b. On Earth’s surface, the atmospheric pressure is approximately 10.13 newtonscm 2.

c. The atmospheric pressure decreases as you rise above Earth’s surface. 13. The depth is 2.5 m.

15. Normal conversation emits 65 decibels.

baseball game increased by 22.5 min.

11. The pH of milk is 6.4. 17. The average time of a Major League

19. The thickness of the copper is 0.4 cm.

21. The Richter scale magnitude of

23. The intensity of the earthquake was 794,328,235I0.

the earthquake was 6.8.

earthquake for the seismogram given is 5.3.

25. The magnitude of the

27. The magnitude of the earthquake for the seismogram given is 5.6.

29. The investment has a value of $3210.06 after 5 years.

CHAPTER 12 REVIEW EXERCISES 2. 52 25 [12.2A]

3.

[12.1B]

y

4. 4

2 –4

5.

2 4 log3 x log3 y 5 5

10. 32 [12.2A]

11. 79 [12.4B]

15. log232 5 [12.2A] 20. 2 [12.4B]

6. log3

[12.2B]

x2 y5

–2

0

–2

4

–4

[12.1B]

7. 12 [12.4A]

17. 0.535 [12.4A] 22.

18.

–4

–2

0

–2

–2

–4

–4

2

4

x

2

[12.4B]

3 2

14.

[12.1A]

[12.3A]

y

2

x

1 4

3

2 4

8.

13. log7xy4 [12.2B]

4

2

0

–4

4

0

–2

–2

12. 1000 [12.2A]

y

–4

2

–4

16. 0.4278 [12.2C]

21.

2

x

–2

[12.2B]

[12.3A]

y

4

4

x

9. 1.7251 [12.2C] 1 5 log8 x 3 log8 y 2

19.

2 3

[12.4B]

23. 2 [12.2A]

[12.2B]


1. 1 [12.1A]

A37


24. log2

3

x y

[12.2B]

25. 1 [12.4A]

26. 1 [12.1A]

1 2 log5 x log5 y [12.2B] 3 3

27.

1 5

28. log381 4 [12.2A]

[12.2A]

31. 2 [12.4A]

1 7

29. 5 [12.4B]

30.

34. 2 [12.4B]

35. 2.3219 [12.2C]

36. 3 [12.4A]

39. 5 [12.1A]

40. 3.2091 [12.4A]

41. The value of the investment in 2 years is $4692. [12.5A]

32.

[12.1A]

37. 625 [12.2A]

42. The Richter scale magnitude of the earthquake is 8.3. [12.5A]

33. 4 [12.2A]

38. logb

x2 y7

[12.2B]

43. The half-life is 27 days. [12.5A]

44. The sound emitted from a busy street corner is 107 decibels. [12.5A]

CHAPTER 12 TEST 1. f 0 1 [12.1A]

2. f 2

1 3

[12.1A]

3.

[12.1B]

y

4. 4

2 –4

5. 2 [12.2A]

6.

1 9

[12.2A]

7.

1 log6 x 3 log6 y 2

13. 2 [12.4A]

10. log3

14. 3 [12.4A]

0

2

4

–4

–4

–4

8. 2

x

–4

–2

0

–2

–2

–4

–4

x y

11. ln x

[12.2B]

15. 2.5789 [12.4A]

19. 2.6801 [12.2C]

1 ln z 2

2

4

x

x

4

[12.2B]

16. 6 [12.4B]

2

[12.3A]

y

2 4

0 –2

4

2

–2

–2

4

0

[12.2B]

[12.2C]

18. 1.3652

–2

2

x

[12.3A]

y

–4

9.

–2

[12.1B]

y

4

12. ln

x3 yz

[12.2B]

17. 3 [12.4B]

20. The half-life is 33 h. [12.5A]


8 7

2. y 2x 6 [4.6A]

[2.2C]

6. 2 10 and 2 10

[10.2A]

3. 4xn 3 xn 1 [7.3A/7.3B] 7. 3x2 8x 3 0

[10.1B]

4.

x3 x3

8.

[8.3A]

5.

xy yx xy

[9.2D]

[5.4A]

y 4


2 –4

–2

0

2

x

4

–2 –4

9. 0, 1, 2 13.

10.

[5.2B]

2x2 17x 13 x 2 2x 3

[12.1B]

y

14.

0

[10.5A]

12. x 1 x 4 [2.6B]

15. 4 [12.1A]

4

2 –2

[12.3A]

y

4

–4

11. x 5 x 1

[8.2B]

2 2

4

x

–4

–2

0

–2

–2

–4

–4

16. 125 [12.2A]

17. log b

x3 y5

[12.2B]

2

4

x

18. 1.7712 [12.2C]

19. 2 [12.4A]

20.

1 2

[12.4B]

A38

Final Exam

21. The customer can write at most 49 checks. [2.5C] 23. The rate of the wind is 25 mph. [8.7B]

22. The cost per pound of the mixture is $3.10. [2.4A]

24. The spring will stretch 10.2 in. [8.8A]

is $.40 per foot. The cost of fir is $.25 per foot. [5.3B]

25. The cost of redwood

26. In approximately 10 years, the investment will be

worth $10,000. [12.5A]

Answers to Final Exam 1. 31 [1.3A]

2. 1 [1.4A]

268.1 ft3. [3.3A]

3. 10x 33 [1.4D]

y

7.

5. 4,

4. 8 [2.2A]

[2.6A]

6. The volume is

2 1 9. y x 3 3

8. y 3x 7 [4.5B]

[4.3C]

2 3

[4.6A]

4 2 –4 –2 0 –2

2

x

4

–4

10. 6a3 5a2 10a [6.3A] 5 13. x2 2x 3 2x 3

22.

7 4

18. d

[8.4A]

x22y 2y2

[6.4B]

[9.2D]

[8.1C]

[8.6A]

y4 162x3

6 3 i [9.4D] 5 5

23.

26. 8, 27 [10.4A]

an a1 n1

x(x 1) 14. 2x 5 19.

[6.1B]

28.

[11.2A]

2

4

x

–4

33. x x

3 2

3 x 6 [11.4B] 2

[4.7A]

0

2

[12.1B]

4

[9.1C]

x

–4

–2

0

41. The cyclist rode 40 mi. [2.4C]

2

4

2

4

x

–4

42. There is $8000 invested at 8.5% and $4000 invested at 6.4%. [5.1C] 44. An additional 200 shares are needed. [8.5B]

46. The object has fallen 88 ft when the speed reaches 75 ft/s. [9.3B]

of the plane for the first 360 mi was 120 mph. [8.7B]

48. The intensity is 200 foot-candles. [8.8A]

the boat in calm water is 12.5 mph. The rate of the current is 2.5 mph. [5.3A] 2 years is $4785.65. [12.5A]

–4 –2 0 –2

40. The range of scores is 69 x 100. [2.5C]

43. The length is 20 ft. The width is 7 ft. [10.6A] the plane was 420 mph. [8.7B]

[12.3A]

2

x

–4

39. 6 [12.4B]

y 4

–2

[12.2B]

x

37.

2 2

4

32. 7x3

4

–4

a2 b2

–2

31. (3, 4) [5.2A]

y

36.

2

38. log2

[4.3A]

–4

4

–4 –2 0 –2

[10.3A]

34. {x4 x 1} [2.6B]

[2.5B]

y

35.

3 17 3 17 , 4 4

–2

–4

30. f 1(x)

[8.3A]

2

–4 –2 0 –2

[10.4C]

25.

x3 x1

21. 2x2y2y [9.2B]

[9.1A]

y

2

3 2

16.

4

4

29. 2,

[8.2B]

1 64x8y5

20.

24. 2x2 3x 2 0 [10.1B]

y

27.

12. (x y)(1 x)(1 x) [7.4D]

10x 15. (x 2)(x 3)

45. The rate of 47. The rate 49. The rate of

50. The value of the investment after


17.

11. (2 xy)(4 2xy x2y2) [7.4B]


A39

Answers to Chapter R Selected Exercises SECTION R.1 1. 10

5. 2

3. 21

25. 8x

27. 3b

45. x

29. 5a

47. 7x

11. 1

51. 3a 15

61. 5x 75

13. 1

33. 15x 2y

31. 11x

49. 3y

59. 8y 7

9. 3

7. 5

17. 6

15. 18

37. 22y 2

35. 9a

55. 10y2 5

53. 16x 48

21. 40

19. 2

1 39. x 2

41. 20x

23. 2.01 43. 24a

57. 18x 2 12xy 6y2

63. a 56

SECTION R.2 1. 12 25.

3. 1

1 2

5. 8 29.

27. 5

41. {x x 2}

14 55. x x 11

7. 20

1 9

31. {x x 3}

43. {x x 3}

11. 1

9. 8

3 57. x x 8

13. 1

33. {n n 2}

45. {x x 2}

5 59. x x 4

3 4

15.

17. 3

35. {x x 3}

47. {x x 2}

19. 3

37. {n n 0}

49. {x x 3}

23. 1

21. 2

39. {x x 4}

51. {x x 2}

53. {x x 4}

SECTION R.3 1.

3.

y 4

(–4, 1) –4

x

–4

–2 –4

y

9.

–4

–2


–4

0

–2

–2

–4

–4

y

2

x

4

–4

0

4

2

2

2

2

x

4

–4

–2

0

2

x

4

–4

–2

0

2

x

4

–4

–2

0

–2

–2

–2

–2

–4

–4

–4

–4

y

–2

y

19.

y

21.

4

4

4

2

2

2

2

0

4

x

–4

–2

0

2

4

x

–4

–2

0

–2

–2

–2

–4

–4

–4

29. Undefined

31. 0

33. 1

35. m

5 , b (0, 4) 2

2

4

x

y

15.

4

2

4

–4

4

0

2

–2

y

13.

–2

4

17.

–4

x

4

–2

11.

2

2

(0, 0) 0 2

–2

(–3, –2)

4

4

2

(4, 0) 2 4

y

7.

y

(3, 4)

4

(2, 3)

2 0

–2

(–2, –2)

5.

y

(–2, 5)

23. 3

2

4

25.

1 4

x

27.

3 2

x

37. m 1, b 0, 0

y

39. 4 2 –4

–2

0

2

4

x

–2 –4

y

41.

y

43.

4 2 –4

–2

0

4

2 2

4

x

–4

–2

0

47. 13; (4, 13)

y

45.

4

2 2

4

x

–4

–2

0

–2

–2

–2

–4

–4

–4

2

4

x

49. 14; (1, 14)

A40

Chapter R

51. 1; (2, 1) 63. y

53. 15; (2, 15)

55. 2; 1, 2

57. y 3x 1

59. y 2x 3

61. y

3 x 5

2 x4 3

SECTION R.4 3. x 15

19. 54n 14 39.

5. x 12y18

21.

1 2x 3

41.

7xz 8y3

23. 8x 13y14

1 2x 2y6

55. 2x 3 x 2 2

9. m 9n 3

7. a 6

43.

1 x6

45.

4 x7

25. y8 x4

11. 8a 9b 3c6

47.

57. x 3 2x 2 6x 6

27. b 10 4a 10

65. 12b 48b 24b

71. x 3 4x 2 5x 2

73. a 3 6a 2 13a 12

2

4

79. y 4y y 5y 2 4

3

2

89. 4x 2 31x 21

99. 10a 2 14ab 12b 2 109. 3y 5 119. x 2 5

81. a a 12

111. b 5

121. y(12y 5)

129. x 2y2(x 2y2 3xy 6) 137. (a 1)(a 2)

49. 7b 2 b 4

5

93. 21a 2 83a 80 103. x 2

131. 8x2y(2 xy3 6y)


157. (2t 1)(5t 3)

163. (t 2)(2t 5)

165. (3y 1)(4y 5)

171. 3(x 2)(x 3)

173. a(b 8)(b 1)

179. 5(t 2)(2t 5)

181. p(2p 1)(3p 1)

9 x 2y 4

37.

2 x4

53. 7x 7

69. 2x 4y 6x 3y2 4x 2y3 77. x 4 4x 3 3x 2 14x 8

85. 2x 15x 7 95. 2x 2 3xy y2

115. 5y 3

133. (x 2)(x 1)

87. 3x 2 11x 4 97. 3x 2 10xy 8y2

107. x 2

125. 5(x 2 3x 7)

149. (x 7)(x 8)

155. (x 3)(2x 1)

35.

2

64 x4

141. (z 5)(z 9)

x6 y12

17. 27m 2n 4p 3

61. y3 y2 6y 6

105. 2y 7

113. 3x 17

123. 5xyz(2xz 3y2)

139. (b 8)(b 1)

33.

75. 2b 3 7b 2 19b 20 2

24 b3

x2 3

2a 3

51. 3a 2 3a 17

3

83. y 10y 21

2

101. 16x 2 49

20 2y 4

31.

67. 9b 9b 24b

3

91. 3y2 2y 16

1 x5

29.

59. 5x 3 10x 2 x 4

63. 4y 2y 2y 4 3

1 d 10

15.

13. mn 2

8 x2

1 2y 3

117. x 2 5x 2

127. 3y2( y2 3y 2) 135. (a 4)(a 3)

143. (z 5)(z 9)

151. ( y 3)(2y 1)

145. (b 4)(b 5)

153. (a 1)(3a 1)

159. (2z 1)(5z 4)


167. (a 5)(11a 1)

169. (2b 3)(3b 2)

175. 2y2( y 3)( y 16)

177. x(x 5)(2x 1)


1. z8

Glossary abscissa The first number of an ordered pair; it measures a horizontal distance and is also called the first coordinate of an ordered pair. [4.1]

circle Plane figure in which all points are the same distance from its center. [3.2, 11.5 – online]

absolute value of a number The distance of the number from zero on the number line. [1.1, 2.6]

clearing denominators Removing denominators from an equation that contains fractions by multiplying each side of the equation by the LCM of the denominators. [2.2]

absolute value inequality An inequality that contains the absolute-value symbol. [2.6] absolute value equation An equation that contains the absolute-value symbol. [2.6] acute angle An angle whose measure is between 0˚ and 90˚. [3.1] acute triangle A triangle that has three acute angles. [3.2] addend In addition, a number being added. [1.1] addition method An algebraic method of finding an exact solution of a system of linear equations. [5.2] additive inverse of a polynomial The polynomial with the sign of every term changed. [6.2] additive inverses Numbers that are the same distance from zero on the number line but lie on different sides of zero; also called opposites. [1.1, 1.4] adjacent angles Two angles that share a common side. [3.1] alternate exterior angles Two nonadjacent angles that are on opposite sides of the transversal and lie outside the parallel lines. [3.1] alternate interior angles Two nonadjacent angles that are on opposite sides of the transversal and between the parallel lines. [3.1] analytic geometry Geometry in which a coordinate system is used to study relationships between variables. [4.1] angle Figure formed when two rays start from the same point. [3.1] area A measure of the amount of surface in a region. [3.2] asymptotes The two straight lines that a hyperbola “approaches.” [11.5 – online]


axes The two number lines that form a rectangular coordinate system; also called coordinate axes. [4.1]

circumference The perimeter of a circle. [3.2]

coefficient The number part of a variable term. [1.4] cofactor of an element of a matrix 1i+j times the minor of that element, where i is the row number of the element and j is its column number. [5.5 – online] combined variation A variation in which two or more types of variation occur at the same time. [8.8] combining like terms Using the Distributive Property to add the coefficients of like variable terms; adding like terms of a variable expression. [1.4] common logarithms Logarithms to the base 10. [12.2] common monomial factor A monomial factor that is a factor of the terms in a polynomial. [7.1] complementary angles Two angles whose measures have the sum 90˚. [3.1] completing the square Adding to a binomial the constant term that makes it a perfect-square trinomial. [10.2] complex fraction A fraction whose numerator or denominator contains one or more fractions. [8.3] complex number A number of the form a bi, where a and b are real numbers and i 1. [9.4] composition of two functions The operation on two functions f and g denoted by f g. The value of the composition of f and g is given by f gx f gx. [11.3] compound inequality Two inequalities joined with a connective word such as and or or. [2.5] compound interest Interest that is computed not only on the original principal but also on the interest already earned. [12.5] conic section A curve that can be constructed from the intersection of a plane and a right circular cone. The four conic sections are the parabola, hyperbola, ellipse, and circle. [11.5 – online]

axis of symmetry of a parabola A line of symmetry that passes through the vertex of the parabola and is parallel to the y-axis for an equation of the form y ax2 bx c or parallel to the x-axis for an equation of the form x ay2 by c. [11.2, 11.5 – online]

conjugates Binomial expressions that differ only in the sign of a term. The expressions a b and a b are conjugates. [9.2]

base In an exponential expression, the number that is taken as a factor as many times as indicated by the exponent. [1.2]

consecutive integers Integers that follow one another in order. [2.3]

basic percent equation Percent times base equals amount. [2.1] binomial A polynomial of two terms. [6.2]

consecutive even integers Even integers that follow one another in order. [2.3]

consecutive odd integers Odd integers that follow one another in order. [2.3] constant function A function given by fx b, where b is a constant. Its graph is a horizontal line passing through (0, b). [4.3]

center of a circle The central point that is equidistant from all the points that make up a circle. [3.2, 11.5 – online]

constant of proportionality k in a variation equation; also called the constant of variation. [8.8]

center of an ellipse The intersection of the two axes of symmetry of the ellipse. [11.5 – online]

constant of variation k in a variation equation; also called the constant of proportionality. [8.8]

G1

Glossary

constant term A term that includes no variable part; also called a constant. [1.4]

empty set The set that contains no elements; also called the null set. [1.5]

coordinate axes The two number lines that form a rectangular coordinate system; also called axes. [4.1]

equation A statement of the equality of two mathematical expressions. [2.1]

coordinates of a point The numbers in the ordered pair that is associated with the point. [4.1]

equilateral triangle A triangle in which all three sides are of equal length. [3.2]

corresponding angles Two angles that are on the same side of the transversal and are both acute angles or are both obtuse angles. [3.1]

equivalent equations Equations that have the same solution. [2.1]

cube A rectangular solid in which all six faces are squares. [3.3] cube root of a perfect cube One of the three equal factors of the perfect cube. [7.4] cubic function A third-degree polynomial function. [6.2] decimal notation Notation in which a number consists of a whole-number part, a decimal point, and a decimal part. [1.2] degree A unit used to measure angles. [3.1]

evaluating a function Replacing x in f(x) with some value and then simplifying the numerical expression that results. [4.2] evaluating a variable expression Replacing each variable by its value and then simplifying the resulting numerical expression. [1.4] even integer An integer that is divisible by 2. [2.3] expanding by cofactors A technique for finding the value of a 3 3 or larger determinant. [5.5 – online]

degree of a monomial The sum of the exponents of the variables. [6.1]

exponent In an exponential expression, the raised number that indicates how many times the factor, or base, occurs in the multiplication. [1.2]

degree of a polynomial The greatest of the degrees of any of the polynomial’s terms. [6.2]

exponential equation An equation in which the variable occurs in the exponent. [12.4]

dependent system of equations A system of equations whose graphs coincide. [5.1]

exponential form The expression 26 is in exponential form. Compare factored form. [1.2]

dependent variable In a function, the variable whose value depends on the value of another variable known as the independent variable. [4.2]

exponential function The exponential function with base b is defined by f(x) bx, where b is a positive real number not equal to one. [12.1]

descending order The terms of a polynomial in one variable are arranged in descending order when the exponents of the variable decrease from left to right. [6.2]

exterior angle An angle adjacent to an interior angle of a triangle. [3.1]

determinant A number associated with a square matrix. [5.5 – online] diameter of a circle A line segment with endpoints on the circle and going through the center. [3.2] diameter of a sphere A line segment with endpoints on the sphere and going through the center. [3.3] difference of two perfect squares A polynomial in the form of a2 b2. [7.4]

extraneous solution When each side of an equation is raised to an even power, the resulting equation may have a solution that is not a solution of the original equation. Such a solution is called an extraneous solution. [9.3] factor In multiplication, a number being multiplied. [1.1] factored form The multiplication 2 2 2 2 2 2 is in factored form. Compare exponential form. [1.2] factoring a polynomial Writing the polynomial as a product of other polynomials. [7.1]

difference of two perfect cubes A polynomial in the form a3 b3. [7.4]

factoring a quadratic trinomial Expressing the trinomial as the product of two binomials. [7.2]

direct variation A special function that can be expressed as the equation y kx, where k is a constant called the constant of variation or the constant of proportionality. [8.8]

FOIL A method of finding the product of two binomials. The letters stand for First, Outer, Inner, and Last. [6.3]

discriminant For an equation of the form ax2 bx c 0, the quantity b2 4ac is called the discriminant. [10.3]

function A relation in which no two ordered pairs that have the same first coordinate have different second coordinates. [4.2]

domain The set of the first coordinates of all the ordered pairs of a relation. [4.2] double root When a quadratic equation has two solutions that are the same number, the solution is called a double root of the equation. [10.1] element of a matrix A number in a matrix. [5.5 – online]

formula A literal equation that states a rule about measurement. [8.6]

functional notation A function designated by f(x), which is the value of the function at x. [4.2] graph of an integer A heavy dot directly above the number on the number line. [1.1]

elements of a set The objects in the set. [1.1, 1.5]

graph of a function A graph of the ordered pairs that belong to the function. [4.3]

ellipse An oval shape that is one of the conic sections. [11.5 – online]

graph of an ordered pair The dot drawn at the coordinates of the point in the plane. [4.1]


G2

Glossary

graphing a point in the plane Placing a dot at the location given by the ordered pair; also called plotting a point in the plane. [4.1]

isosceles triangle A triangle that has two sides of equal length; the angles opposite the equal sides are of equal measure. [3.2]

greater than A number that lies to the right of another number on the number line is said to be greater than that number. [1.1]

joint variation A variation in which a variable varies directly as the product of two or more variables. A joint variation can be expressed as the equation z kxy, where k is a constant. [8.8]

greatest common factor (GCF) The greatest common factor of two or more integers is the greatest integer that is a factor of all the integers. [7.1] greatest common factor (GCF) of two or more monomials The greatest common factor of two or more monomials is the product of the GCF of the coefficients and the common variable factors. [7.1] grouping symbols Parentheses ( ), brackets [ ], braces { }, the absolute value symbol, and the fraction bar. [1.3] half-plane The solution set of a linear inequality in two variables. [4.7] horizontal-line test A graph of a function represents the graph of a one-to-one function if any horizontal line intersects the graph at no more than one point. [11.4] hyperbola A conic section formed by the intersection of a cone and a plane perpendicular to the base of the cone. [11.5 – online]

leading coefficient In a polynomial, the coefficient of the variable with the largest exponent. [6.2] least common denominator The smallest number that is a multiple of each denominator in question. [1.2] least common multiple (LCM) The LCM of two or more numbers is the smallest number that is a multiple of each of those numbers. [1.2] least common multiple (LCM) of two or more polynomials The simplest polynomial of least degree that contains the factors of each polynomial. [8.2] leg In a right triangle, one of the two sides that are not opposite the 90˚ angle. [9.3] less than A number that lies to the left of another number on the number line is said to be less than that number. [1.1]

hypotenuse In a right triangle, the side opposite the 90˚ angle. [9.3]

like terms Terms of a variable expression that have the same variable part. Having no variable part, constant terms are like terms. [1.4]

imaginary number A number of the form ai, where a is a real number and i 1. [9.4]

line Having no width, it extends indefinitely in two directions in a plane. [3.1]

imaginary part of a complex number For the complex number a bi, b is the imaginary part. [9.4]

linear equation in three variables An equation of the form Ax By Cz D where A, B, and C are coefficients of the variables and D is a constant. [5.2]

inconsistent system of equations A system of equations that has no solution. [5.1] independent system of equations A system of equations whose graphs intersect at only one point. [5.1] independent variable In a function, the variable that varies independently and whose value determines the value of the dependent variable. [4.2] n

index In the expression a, n is the index of the radical. [9.1]


G3

linear equation in two variables An equation of the form y mx b, where m is the coefficient of x and b is a constant; also called a linear function. [4.3] linear function A function that can be expressed in the form y mx b. Its graph is a straight line. [4.3, 6.2] linear inequality in two variables An inequality of the form y mx b or Ax By C. The symbol could be replaced by , , or . [4.7]

inequality An expression that contains the symbol , , (is greater than or equal to), or (is less than or equal to). [1.5]

line segment Part of a line that has two endpoints. [3.1]

integers The numbers . . . , 3, 2, 1, 0, 1, 2, 3, . . . . [1.1]

logarithm For b greater than zero and not equal to 1, the statement y logb x (the logarithm of x to the base b) is equivalent to x b y. [12.2]

interior angle of a triangle Angle within the region enclosed by a triangle. [3.1] intersecting lines Lines that cross at a point in the plane. [3.1] intersection of two sets The set that contains all elements that are common to both of the sets. [1.5]

literal equation An equation that contains more than one variable. [8.6]

lower limit In a tolerance, the lowest acceptable value. [2.6] main fraction bar The fraction bar that is placed between the numerator and denominator of a complex fraction. [8.3]

inverse of a function The set of ordered pairs formed by reversing the coordinates of each ordered pair of the function. [11.4]

matrix A rectangular array of numbers. [5.5 – online]

inverse variation A function that can be expressed as the k equation y , where k is a constant. [8.8] x

minimum value of a function The least value that the function can take on. [11.1]

irrational number The decimal representation of an irrational number never terminates or repeats and can only be approximated. [1.2, 9.2]

maximum value of a function The greatest value that the function can take on. [11.1]

minor of an element The minor of an element in a 3 3 determinant is the 2 2 determinant obtained by eliminating the row and column that contain that element. [5.5 – online]

Glossary

monomial A number, a variable, or a product of a number and variables; a polynomial of one term. [6.1, 6.2]

origin The point of intersection of the two number lines that form a rectangular coordinate system. [4.1]

multiplication The process of finding the product of two numbers. [1.1]

parabola The graph of a quadratic function is called a parabola. [11.1]

multiplicative inverse The multiplicative inverse of a 1 nonzero real number a is ; also called the reciprocal. a [1.4]

parallel lines Lines that never meet; the distance between them is always the same. In a rectangular coordinate system, parallel lines have the same slope and thus do not intersect. [3.1, 4.6]

natural exponential function The function defined by fx e x, where e 2.71828. [12.1]

parallelogram Four-sided plane figure with opposite sides parallel. [3.2]

natural logarithm When e (the base of the natural exponential function) is used as the base of a logarithm, the logarithm is referred to as the natural logarithm and is abbreviated ln x. [12.2]

percent Parts of 100. [1.2]

natural numbers The numbers 1, 2, 3. . . . ; also called the positive integers. [1.1] negative integers The numbers. . . , 3, 2, 1. [1.1] negative reciprocal The negative reciprocal of a nonzero 1 real number a is . [11.1] a negative slope The slope of a line that slants downward to the right. [4.4]

perfect cube The product of the same three factors. [7.4] perfect square The product of a term and itself. [1.2, 7.4] perfect-square trinomial The square of a binomial. [7.4] perimeter The distance around a plane geometric figure. [3.2] perpendicular lines Intersecting lines that form right angles. The slopes of perpendicular lines are negative reciprocals of each other. [3.1, 4.6] plane A flat surface that extends indefinitely. [3.1] plane figure A figure that lies entirely in a plane. [3.1]

nonfactorable over the integers A polynomial is nonfactorable over the integers if it does not factor using only integers. [7.2]

plotting a point in the plane Placing a dot at the location given by the ordered pair; also called graphing a point in the plane. [4.1]

nth root of a A number b such that bn a. The nth root of a can be written a1/n or a. [9.1]

point-slope formula The equation y y1 m(x x1, where m is the slope of a line and x1, y1 is a point on the line. [4.5]

null set The set that contains no elements; also called the empty set. [1.5]

polygon A closed figure determined by three or more line segments that lie in a plane. [3.2]

numerical coefficient The number part of a variable term. When the numerical coefficient is 1 or 1, the 1 is usually not written. [1.4]

polynomial A variable expression in which the terms are monomials. [6.2]

obtuse angle An angle whose measure is between 90˚ and 180˚. [3.1]

positive integers The numbers 1, 2, 3. . . . ; also called the natural numbers. [1.1]

obtuse triangle A triangle that has one obtuse angle. [3.2]

positive slope The slope of a line that slants upward to the right. [4.4]

odd integer An integer that is not divisible by 2. [2.3] one-to-one function In a one-to-one function, given any y, there is only one x that can be paired with the given y. [11.4] opposites Numbers that are the same distance from zero on the number line but lie on different sides of zero; also called additive inverses. [1.1] order m n A matrix of m rows and n columns is of order m n. [5.5 – online] Order of Operations Agreement A set of rules that tells us in what order to perform the operations that occur in a numerical expression. [1.3]

prime polynomial A polynomial that is nonfactorable over the integers. [7.2] principal square root The positive square root of a number. [1.2, 9.1] product In multiplication, the result of multiplying two numbers. [1.1] product of the sum and difference of two terms A polynomial that can be expressed in the form a ba b. [7.4] proportion An equation that states the equality of two ratios or rates. [8.4]

ordered pair A pair of numbers expressed in the form (a, b) and used to locate a point in the plane determined by a rectangular coordinate system. [4.1]

Pythagorean Theorem The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs. [9.3]

ordered triple Three numbers expressed in the form (x, y, z) and used to locate a point in the xyz-coordinate system. [5.2]

quadrant One of the four regions into which a rectangular coordinate system divides the plane. [4.1]

ordinate The second number of an ordered pair; it measures a vertical distance and is also called the second coordinate of an ordered pair. [4.1]

quadratic equation An equation of the form ax2 bx c 0, where a and b are coefficients, c is a constant, and a 0; also called a second-degree equation. [7.5, 10.1]


G4

Glossary

quadratic equation in standard form A quadratic equation written in descending order and set equal to zero. [7.5]

reciprocal of a rational expression The rational expression with the numerator and denominator interchanged. [8.1]

quadratic formula A general formula, derived by applying the method of completing the square to the standard form of a quadratic equation, used to solve quadratic equations. [10.3]

rectangle A parallelogram that has four right angles. [3.2]

quadratic function A function that can be expressed by the equation f(x) ax2 bx c, where a is not equal to zero. [6.2, 11.1]

rectangular solid A solid in which all six faces are rectangles. [3.3]

quadratic inequality An inequality that can be written in the form ax2 bx c 0 or ax2 bx c 0, where a is not equal to zero. The symbols and can also be used. [10.5]

relation A set of ordered pairs. [4.2]

quadratic trinomial A trinomial of the form ax bx c, where a and b are nonzero coefficients and c is a nonzero constant. [7.4] 2

quadrilateral A four-sided closed figure. [3.2] quotient In division, the result of dividing the divisor into the dividend. [1.1] radical equation An equation that contains a variable expression in a radicand. [9.3] radical sign The symbol , which is used to indicate the positive, or principal, square root of a number. [1.2, 9.1] radicand In a radical expression, the expression under the radical sign. [1.2, 9.1] radius of a circle Line segment from the center of the circle to a point on the circle. [3.2, 11.5 – online] radius of a sphere A line segment going from the center to a point on the sphere. [3.3] range The set of the second coordinates of all the ordered pairs of a relation. [4.2] rate The quotient of two quantities that have different units. [8.5] rate of work That part of a task that is completed in one unit of time. [8.7] ratio The quotient of two quantities that have the same unit. [8.5] rational expression A fraction in which the numerator or denominator is a polynomial. [8.1] Copyright © Houghton Mifflin Company. All rights reserved.

G5

a rational number A number of the form , where a and b b are integers and b is not equal to zero. [1.2] rationalizing the denominator The procedure used to remove a radical from the denominator of a fraction. [9.2] ray Line that starts at a point and extends indefinitely in one direction. [3.1]

rectangular coordinate system A coordinate system formed by two number lines, one horizontal and one vertical, that intersect at the zero point of each line. [4.1]

regular polygon A polygon in which each side has the same length and each angle has the same measure. [3.2] repeating decimal A decimal that is formed when dividing the numerator of its fractional counterpart by the denominator results in a decimal part wherein a block of digits repeats infinitely. [1.2] right angle An angle whose measure is 90˚. [3.1] right triangle A triangle that contains a 90˚ angle. [3.1] roster method A method of designating a set by enclosing a list of its elements in braces. [1.5] scalene triangle A triangle that has no sides of equal length; no two of its angles are of equal measure. [3.2] scatter diagram A graph of collected data as points in a coordinate system. [4.1] scientific notation Notation in which a number is expressed as the product of a number between 1 and 10 and a power of 10. [6.1] second-degree equation An equation of the form ax2 bx c 0, where a and b are coefficients, c is a constant, and a 0; also called a quadratic equation. [10.1] set A collection of objects. [1.1, 1.5] set-builder notation A method of designating a set that makes use of a variable and a certain property that only elements of that set possess. [1.5] similar objects Similar objects have the same shape but not necessarily the same size. [8.5] simplest form of a rational expression A rational expression is in simplest form when the numerator and denominator have no common factors. [8.1] slope A measure of the slant, or tilt, of a line. The symbol for slope is m. [4.4] slope-intercept form of a straight line The equation y mx b, where m is the slope of the line and (0, b) is the y-intercept. [4.4] solution of a system of equations in three variables An ordered triple that is a solution of each equation of the system. [5.2]

real numbers The rational numbers and the irrational numbers taken together. [1.2]

solution of a system of equations in two variables An ordered pair that is a solution of each equation of the system. [5.1]

real part of a complex number For the complex number a bi, a is the real part. [9.4]

solution of an equation A number that, when substituted for the variable, results in a true equation. [2.1]

1 reciprocal The reciprocal of a nonzero real number a is ; a also called the multiplicative inverse. [1.2, 1.4]

solution of an equation in three variables An ordered triple (x, y, z) whose coordinates make the equation a true statement. [5.2]

Glossary

solution of an equation in two variables An ordered pair whose coordinates make the equation a true statement. [4.1]

transversal A line that intersects two other lines at two different points. [3.1]

solution set of a system of inequalities The intersection of the solution sets of the individual inequalities. [5.4]

trinomial A polynomial of three terms. [6.2]

solution set of an inequality A set of numbers, each element of which, when substituted for the variable, results in a true inequality. [2.5] solving an equation Finding a solution of the equation. [2.1] sphere A solid in which all points are the same distance from point O, which is called the center of the sphere. [3.3] square A rectangle with four equal sides. [3.2] square of a binomial A polynomial that can be expressed in the form a b2. [6.3] square root A square root of a positive number x is a number a for which a2 x. [1.2] square matrix A matrix that has the same number of rows as columns. [5.5 – online] square root of a perfect square One of the two equal factors of the perfect square. [7.4] standard form of a quadratic equation A quadratic equation is in standard form when the polynomial is in descending order and equal to zero. [7.5, 10.1] substitution method An algebraic method of finding an exact solution of a system of linear equations. [5.1] straight angle An angle whose measure is 180˚. [3.1] sum In addition, the total of two or more numbers. [1.1] supplementary angles Two angles whose measures have the sum 180˚. [3.1] synthetic division A shorter method of dividing a polynomial by a binomial of the form x a. This method uses only the coefficients of the variable terms. [6.4] system of equations Two or more equations considered together. [5.1] system of inequalities Two or more inequalities considered together. [5.4] terminating decimal A decimal that is formed when dividing the numerator of its fractional counterpart by the denominator results in a remainder of zero. [1.2] terms of a variable expression The addends of the expression. [1.4] tolerance of a component The amount by which it is acceptable for the component to vary from a given measurement. [2.6]

triangle A three-sided closed figure. [3.1] undefined slope The slope of a vertical line is undefined. [4.4] uniform motion The motion of an object whose speed and direction do not change. [2.1] union of two sets The set that contains all elements that belong to either of the sets. [1.5] upper limit In a tolerance, the greatest acceptable value. [2.6] value of a function The value of the dependent variable for a given value of the independent variable. [4.2] variable A letter of the alphabet used to stand for a number that is unknown or that can change. [1.1] variable expression An expression that contains one or more variables. [1.4] variable part In a variable term, the variable or variables and their exponents. [1.4] variable term A term composed of a numerical coefficient and a variable part. When the numerical coefficient is 1 or 1, the 1 is usually not written. [1.4] vertex Point at which the rays that form an angle meet. [3.1] vertex of a parabola The point on the parabola with the smallest y-coordinate or the largest y-coordinate. [11.1] vertical angles Two angles that are on opposite sides of the intersection of two lines. [3.1] vertical-line test A graph defines a function if any vertical line intersects the graph at no more than one point. [11.2] volume A measure of the amount of space inside a closed surface. [3.3] x-coordinate The abscissa in an xy-coordinate system. [4.1] x-intercept The point at which a graph crosses the x-axis. [4.3] y-coordinate The ordinate in an xy-coordinate system. [4.1] y-intercept The point at which a graph crosses the y-axis. [4.3] zero of a function A value of x for which fx 0. [11.1] zero slope The slope of a horizontal line. [4.4]


G6

Index


A Abscissa, 219, 295 Absolute value, 4, 63, 137 applications of, 141–142 and distance, 137 equations with, 137–138, 151 inequalities with, 139–140, 151 for roots, 521 Absolute-value equations, 137–139, 149, 151 Absolute-value function, graph of, 626 Absolute-value inequalities, 139–141, 149, 151 Acute angle, 164, 208 Acute triangle, 178, 209 Addend, 5 Addition, 64–65 of additive inverses, 40 Associative Property of, 39, 66 Commutative Property of, 39, 66 of complex numbers, 544–545, 549, 554 of decimals, 20 Distributive Property of, 38 of fractions, 19–20, 65 of functions, 631 of integers, 5–7 Inverse Property of, 40, 66 of polynomials, 360–362, 364, 391 of radical expressions, 528–529, 533–534 of rational expressions, 461, 463, 466–468, 507 of rational numbers, 19–20 verbal phrases for, 44 Addition method of solving systems of equations, 309–310, 314–315, 317–320 Addition Property of Equations, 74, 92, 150, 485, 716 Addition Property of Inequalities, 125, 151 Addition Property of Zero, 39, 66 Additive inverse, 4, 40, 63 of polynomials, 360–362, 391 Adjacent angles, 164, 166–168, 208 as supplementary angles, 166 Algebraic fraction(s), see Rational expression(s) al Hassar, 17 al-Khowarizmi, 571 Alternate exterior angles, 167, 208 Alternate interior angles, 167, 208 Angle(s), 162, 208 acute, 164, 208 adjacent, 164, 166, 168, 208 alternate exterior, 167, 208 alternate interior, 167, 208 complementary, 163 corresponding, 167, 208, 479 exterior, 169, 170 formed by intersecting lines, 166–168 interior, 169, 170

measure of, 162, 165 obtuse, 164, 208 right, 163 sides of, 162 straight, 164 supplementary, 164 symbol for, 162 of a triangle, 169, 170, 209, 479 vertex of, 162 vertical, 166, 170, 208, 479 Antecedent, 503 Antilogarithm, 670 Apothem, 194 Application problems, 10 business, 648–649 containing radicals, 539–540 exponential functions, 689–692 exponential growth, 689 factoring, 435–436, 593–594 implications, 503 inequalities, 130, 141–142 integers, 435 investment, 303–304 logarithmic functions, 689–692 mixtures, 113–116 multiplication of polynomials, 369–370 physics, 148 Principle of Zero Products, 435–436 proportion, 478 rate of wind or current, 321–322, 491–492, 594 scientific notation, 352 units in, 164 using formulas, 99 using linear functions, 267 using parabolic equations, 616–618 using quadratic equations, 593–594 using right triangles, 540 using similar triangles, 478 using system of equations, 321–324 Approximation of logarithms, 670 of quotients, 21 of radical expressions, 526 of the value of a function, 661 Archimedes, 180 Area(s), 182–186 formulas for, 182–186, 210 surface, 198–200 Associative Property of Addition, 39, 66 Associative Property of Multiplication, 40, 66 Astronomical unit, 388 Average, 10 Axes, 219 Axis of symmetry, 207 parabola, 610, 650, 651

B Base in exponential expression, 22, 345 in exponential function, 670, 700

in logarithmic expression, 670, 700 of parallelogram, 183 in percent equation, 77 of trapezoid, 184 of triangle, 184 Basic percent equation, 77, 87, 150 Binomial(s), 357, 389 factors, 403, 405, 443 product of, 367 square of a, 368, 571 Binomial factor, 403, 405, 443 Brahmagupta, 17 Briggs, Henry, 672 Burgi, Jobst, 672 Business, application problems, 648–649

C Calculator(s), 62 to check solutions of quadratic equation, 573 evaluating expressions with irrational exponents, 661 evaluating expressions with rational exponents, 517 evaluating logarithms, 684 to find minimum or maximum of a function, 648 to find powers, 669 graphing functions, 284 graphing linear equations, 284 natural exponential function on, 662 to solve exponential equations, 698 to solve first-degree equations, 284 to solve logarithmic equations, 698 to solve quadratic equations, 597–598 to solve radical equations, 552 to solve systems of equations, 333–335 Calculus, 255, 616 Candela, 504 Carbon dating, 690 Cardan, Hieronimo, 543 Cartesian coordinate system, 219–220 Center of circle, 180 of sphere, 195 Change-of-Base Formula, 674, 701 Checking factoring, 408–410, 424 solution(s) of an equation, 76, 477, 538, 566, 684 solution(s) of a system of equations, 309, 314, 329 Chord, 283 Chuquet, Nicolas, 517 Circle, 180–181, 209 area of, 185–186, 210 center of, 180 chord, 283 circumference, 180, 210 radius, 180, 209 Circumference, 180–181, 210

I1

Index

Clearing denominators, 92, 149 Coefficient(s), 37, 64 leading, 357, 389 Cofactor(s), 298 Combined variation, 498, 506 Combining like terms, 39, 64 Common denominator, 19, 92–93, 461, 465–466 least common multiple, 19, 92–93, 461–464, 469. 505 Common factor, 401, 409 Common logarithm, 670, 700 Commutative Property of Addition, 39, 66 Commutative Property of Multiplication, 41, 66 Complementary angles, 163–165, 208 Complete factoring, 409–410 Completing the square, 571, 600, 609–610 to find vertex and axis of symmetry of parabola, 610–611 geometric method, 571 quadratic formula derivation, 577–579 to solve quadratic equations, 571–576, 600 Complex fraction(s), 469–472, 505, 507 Complex number(s), 543–548, 553 addition of, 544–545, 549, 554 conjugate of, 548 division of, 548 imaginary part, 543, 553 multiplication of, 545–547, 549–550, 554 real part, 543, 553 simplifying, 543–544, 549 subtraction of, 544–545, 549, 554 Composite function, 633 Composition of inverse functions, 642, 652 of two functions, 633–635, 636–637, 652 Compound inequality, 128, 149 Compound interest formula, 689 Cone, 195 surface area of, 199–200, 210 volume of, 196 Conjugate, 530, 553 of a complex number, 548 Consecutive even integers, 106, 150, 435 Consecutive integers, 106, 149, 150 Consecutive odd integers, 106, 150 Consequent, 503 Constant function, 244, 245 Constant of proportionality, 497, 506 Constant of variation, 497, 506 Constant term, 37, 64, 357, 389, 716 degree of, 345 of a perfect-square trinomial, 571 Contradiction, proof by, 697 Contrapositive, 503

Converse, 503 Coordinate(s), 219, 285 Coordinate axes, 219, 285 Coordinate system, rectangular, 219–220 Corresponding angles, 167, 208, 479 Credit reports, 679 Cube, 196, 209 of a number, 22 surface area of, 198–199, 210 of a variable, 45 volume of, 196, 210 Cube root(s), 403, 425, 522 Cubes, sum and difference of, 425 factoring, 425 Cubic function, 357–359, 389 graphing, 625 Current or rate-of-wind problems, 321–322, 491–492, 594 Cylinder, 195, 209 surface area of, 198, 199–200, 210 volume of, 196, 210

D DaVinci, Leonardo, 179 Decagon, 177 Decimal(s), 17 addition of, 20 and fractions, 17–19 multiplication of, 21–22 notation, 17 operations on, 20, 21 and percents, 18, 65 repeating, 17, 24, 63 representing irrational numbers, 24 representing rational numbers, 17 and scientific notation, 351 subtraction of, 20 terminating, 17, 63 See also Rational numbers Dedekind, Richard, 527 Degree in angle measurement, 162, 208 of a constant term, 345 of a monomial, 345 of a polynomial,. 357, 389 of a quadratic equation, 563 Denominator(s) clearing, 92, 149 common, 19 least common multiple, 19, 92–93, 461–464, 469, 505 rationalization of, 531–532, 553 Dependent system of equations, 298, 313, 335 Dependent variable, 230, 286 Descartes, René, 22 Descending order, 357, 389, 433, 443 Diameter of a circle, 180, 209 of a sphere, 195 Difference of two cubes, 425–426 of two squares, 423

Dimensional analysis, 386 Direct variation, 497, 506 Discriminant, 579, 599, 613–614, 651 Distance and absolute value, 137 motion problems, 117, 321, 491 between two points on the number line, 137, 161, 164–165 Distributive Property, 38, 66 factoring and, 401–402 use in multiplying polynomials, 365–367 use in simplifying expressions, 38–39, 42–43, 64, 717–718 use in solving inequalities, 115 Division, 65 of complex numbers, 548 of decimals, 21 exponential expressions and, 349, 390 of fractions, 21, 65–66 of functions, 631–632 of integers, 8–10 of monomials, 347–350, 353–354 by one, 9, 65 of polynomials, 375–380, 391 of radical expressions, 531–532, 535–536 of rational expressions, 456, 459–460, 507 of rational numbers, 21 and reciprocals, 21 synthetic, 378–381, 383–385, 391 verbal phrases for, 45 by zero, 9, 65 Domain, 230, 233, 286, 641 estimating from a graph, 627–628 of exponential function, 662 of inverse function, 640 of radical function, 626 Double function, 231 Double root, 563, 598

E Einstein, Albert, 73 Element, 3 of a set, 55, 64 Empty set, 55, 64 Endpoint, 161 Equals sign, 74 Equation(s), 73, 149 with absolute value, 137–138, 151 Addition Property of, 74, 92, 150, 485 basic percent, 77, 87, 150 containing parentheses, 92, 103 equivalent, 74 exponential, 683, 701 finding, 264–267 first-degree, 92–99 formulas, see Formula(s) fractional, 92, 473–476, 508, 568 graphs, see Graph(s) linear, see Linear equation(s)


I2


Index

literal, 485–486, 506, 508 logarithmic, 685, 687 Multiplication Property of, 75–76, 92, 150, 485 of parabola, 609, 650 percent mixture, 78 quadratic, see Quadratic equation(s) radical, 537 reducible to quadratic, 584–588 second-degree, 563, 598 simple interest, 78, 150 solution of, 73, 75, 92–95, 149. See also Solving equations solving, see Solving equations systems, see Systems of equations translating into, from sentences, 106–107 in two variables, 221 of variation, 497–501, 506 Equilateral triangle, 177, 209 Equivalent equations, 74, 149 Euclid, 162 Euler, Leonhard, 662 Evaluating expressions absolute value, 4 exponential, 23 logarithmic, 668, 674 numerical, 33–34 variable, 37–38, 64 Evaluating functions, 232, 235, 286 Even integer, 106, 150 Exponent(s), 22, 63 in division, 349–350 irrational, 661 in multiplication, 345 negative, 348, 349, 390 of perfect cubes, 425 of perfect squares, 423 raising a power to a power, 346 rational, 517–518, 523–524 rules of, 345–346 zero as, 347, 390 Exponential decay equation, 690, 693, 701 Exponential equations, 683, 701 solving, 683–684 writing as radical expressions, 519–520, 525, 554 writing equivalent logarithmic equations, 669, 679 Exponential expression(s), 22–23, 345–350, 519 base of, 22, 63, 345 division of, 349 evaluating, 23 factored form of, 22 multiplication of, 345, 386 and radical expressions, 519, 525, 554 as repeated multiplication, 22 simplifying, 23, 346, 390 writing as radical expressions, 519–520, 525, 554

Exponential form, 22, 63 Exponential function(s), 661–667, 700 applications of, 689–692 evaluating, 661, 665–666 graph of, 663–664, 666 inverse of, 669 natural, 662, 700 range of, 662 Exponential growth equation, 689, 701 Exponential notation, 22 Export, 63 Expression(s) exponential, 22–23, 345–350, 519 like terms of, 38–39 logarithmic, 669 radical, 24 rational, 453, 505 with rational exponents, 517–518, 523–524 terms of, 37 variable, 37 verbal, 44 Exterior angle, 169 Extraneous solution, 538

F Faces of a rectangular solid, 195 Factor(s), 7 binomial, 403–405, 443 common, 19, 401, 409 greatest common (GCF), 401–402, 410, 442 perfect square, 24 prime, 427 trial, 415–416, 419–420 Factored form of an exponential expression, 22 Factoring common factors, 401, 409 completely, 409–410, 427–428, 431–432, 443–444 difference of two cubes, 425, 430, 444 difference of two squares, 423–424, 429–430, 444 by grouping, 403–406, 417–418, 421–422, 444 monomial from polynomial, 401–402 perfect-square trinomial, 423–424, 426, 429–430, 444 polynomials, 407–428, 442 solving equations by, 433–436, 563–564 sum of two cubes, 425–426, 430, 444 trinomials, 407–428 trinomials that are quadratic in form, 426–428, 431 by using trial factors, 415–416, 419–420 Fibonacci, 3, 76

I3

FICO scores, 699 First coordinate, see Abscissa First-degree inequalities, 125–127 in two variables, 279, 286 FOIL method, 367–368, 391, 415 Formula(s), 99, 485, 506 application problems, 99 for area, 182–186, 210 change-of-base, for logarithms, 674 for circumference, 180 compound interest, 689 for distance (rate, time), 117 for perimeter, 178–179 point-slope, 264, 287 quadratic, 577–579, 581–582, 600 slope, 253, 287 for surface area, 198 for volume, 196, 210 Fraction(s), 17, 63 addition of, 19–20, 65 algebraic, see Rational Expression(s) complex, 469–472 and decimals, 17–19 division of, 21, 65–66 multiplication of, 20, 65–66 origin of concept, 17 and percent, 18, 65 reciprocals of, 21 solving equations containing, 92, 473–476 subtraction of, 19–20, 65 See also Rational number(s) Fractional equations, 92, 473–476, 508 reducible to quadratic, 588 Function(s), 229–234, 286 absolute value, 626 addition of, 631 composite, 633–635, 636–637, 652 constant, 244 cubic, 357, 389 domain of, 230, 233–234 double, 231 evaluating, 232, 235, 286 exponential, 661–667, 689–691, 700 graph of, 241–242, 286, 625–630 horizontal line test for, 640, 652, 679 inverse, 640–642, 644–645, 651, 652 linear, 241, 286, 357, 389 logarithmic, 669 one-to-one, 639–640, 643, 651 operations on, 631–632, 636, 652 polynomial, 357–358, 389 quadratic, 357, 389, 650 radical, 626 range of, 230, 233 square, 230–231 subtraction of, 631 value of, 232, 286 vertical-line test for, 626, 652, 679 zeros of, 612–614, 621–622, 650 Functional notation, 231–233, 286

Index

G Gauss, Karl Friedrich, 283 Geometric solid(s), 195–199, 209 Geometry, 151–170, 177–186, 195–200 Graph(s), 286–287, 625–630 of absolute value function, 626 applications of, 248 with calculator, see Calculator(s) of constant function, 287 of cubic functions, 625 of exponential functions, 663–664, 666 of functions, 241–243, 625–630 of horizontal line, 244 of inequalities in one variable, 57–58 of inequalities in two variables, 279, 281–282 of integers, 3 of inverse functions, 641 of linear equation in three variables, 312–315 of linear equation in two variables, 243–248 of linear functions, 241–246 of logarithmic functions, 679–682 of one-to-one functions, 639 of ordered pair, 219–223, 285 of ordered triple, 312 of parabolas, 609–610, 650 of point, 285 of polynomial functions, 358–360, 625 of quadratic functions, 609–611, 619–620, 651 of radical functions, 626 of scatter diagram, 223 of solution sets of systems of inequalities, 329 of systems of linear equations, 297–300 using x- and y-intercepts, 246–247 of vertical line, 244, 245–246 Graphing calculator(s), see Calculator(s) Greater than, 3, 63 Greater than or equal to, 3, 63 Greatest common factor (GCF), 401–402, 410, 442 Grouping, factoring by, 403–406 Grouping symbols, 33–34

H Half-plane, 279, 286 Harriot, Thomas, 57 Height of parallelogram, 183 of trapezoid, 184 of triangle, 184 Hemisphere, 204 Heptagon, 177 Hexagon, 177 Hexagram, 194 Hoppe, Stephen, 314

Horizontal line, 244 Horizontal-line test, 640, 652, 679 Hubble Space Telescope, 388 Hypatia, 162 Hypotenuse, 539

I Illumination, intensity of, 504 Imaginary number (i), 543 Implication(s), 503 Inconsistent system of equations, 298, 313, 335 Independent system of equations, 297, 313, 335 Independent variable, 230, 286 Index of a radical expression, 521 Inductive reasoning, 61 Inequality(ies), 3–4, 57 with absolute value, 139–140, 149, 151 Addition Property of, 125, 151 application problems, 130, 141–142 compound, 128–129, 149 first-degree, 125–128 graphs in one variable, 57–58 graphs in two variables, 279–280, 281–282 linear, 279, 286 Multiplication Property of, 126, 151 nonlinear, 589–592 quadratic, 589, 592 solution set in one variable, 125, 149 solution set in a system of, 313 solution set in two variables, 279–280, 286 solving, see Solving inequalities systems of, 329–332, 336 Inequality symbols, 3 Input, 609, 650 Input/output table, 609, 650 Integer(s), 3, 63, 106 addition of, 5–6 application problems, 435 consecutive, 106, 149, 150 consecutive even, 106, 150, 435 consecutive odd, 106, 150 division of, 8–9 even, 106, 150 graphs of, 3 multiplication of, 7–8 negative, 3 odd, 106 positive, 3 subtraction of, 6 Intercepts of linear equations, 287 of a parabola, 612, 614, 621–622, 651 x- and y-, 246, 286, 612, 650 Interest, 101 compound, 689 simple, 78, 150, 303, 336 Interior angle, 169, 170 Intersecting lines, 162, 168, 207 Intersection of sets, 55, 58, 64

Inverse additive, 4, 40, 63, 360, 391 of exponential function, 669 of functions, 640–642, 644–645, 651 multiplicative, 41, 64 Inverse function, 640–642, 644–645, 652 Inverse Property of Addition, 40, 66 Inverse Property of Logarithms, 671–673, 684, 685, 701 Inverse Property of Multiplication, 41, 66 Inverse variation, 498, 506 Investment problems, 303–304 Irrational number, 24, 63, 527, 670 Isosceles trapezoid, 178 Isosceles triangle, 177, 209

J Joint variation, 498, 506

K Karat, 78

L LCD, see Least common denominator LCM, see Least common multiple Leading coefficient, 357, 389 Least common denominator (LCD), 19 Least common multiple (LCM), 19, 92–93, 461–464, 469, 505 Legs of a right triangle, 539 Leibnitz, Gottfried, 219 Less than, 3, 63 Less than or equal to, 3, 63 Light-year, 388 Like terms, 38–39, 64, 717 Line(s), 161, 207 equations of, see Linear equations horizontal, 244 intersecting, 162, 207 parallel, 162, 168, 207, 273 perpendicular, 163, 207, 274–276 slopes of, 253 slopes of parallel lines, 273–274, 276, 287 slopes of perpendicular lines, 274–275, 287 vertical, 244, 245–246 Linear equation(s), 264–266 applications of, 248, 255–256 graphing, 241–247, 253–258 slope, 253 slope-intercept form, 257–258 solution of, 221 systems of, see Systems of equations in three variables, 312–315 in two variables, 243–248, 257–258 writing, given the slope and a point on a line, 264–265, 268–269 writing, given two points, 265–266, 269–270 x- and y-intercepts of, 246, 286


I4

Index

Linear function(s), 241, 286, 357, 389 applications of, 267 graphs of, 241–246 in two variables, 217 See also Linear equation(s) Linear inequality, 279, 286 system of, 329–332 Line segment, 161, 164–165 Literal equations, 485–486, 506, 508 Logarithm(s), 668–678, 700 antilogarithm, 670 approximation of, 670 change-of-base formula for, 674, 678 common, 670, 700 equations, solving, 685–686, 698 natural, 670, 700 Property of One, 672, 701 Property of Powers, 672–673, 701 Property of Products, 671–673, 701 Property of Quotients, 672–673, 701 use in solving exponential equations, 683 Logarithmic equation(s), 685–688, 698 converting to exponential equations, 669, 679 solving with graphing calculator, 698 Logarithmic function(s), 669 applications of, 689–692 graphs of, 679–682 properties of, 671–673, 684, 685, 701 Lower limit, 141, 149 Lumens, 504

of complex numbers, 545–547, 549–550, 554 of decimals, 21–22 Distributive Property of, 38, 66 of exponential expressions, 345, 386, 389 FOIL method, 367–368 of fractions, 20, 22, 65–66, 432 of integers, 7–10 Inverse Property of, 41, 66 of monomials, 345, 353 of multiplicative inverses, 41, 64 by one, 41, 66 of polynomial and monomial, 365, 371 of polynomials, 365–370, 372, 391 of radical expressions, 529–530, 534–535 of rational expressions, 454–455, 458–459, 506 of rational numbers, 20, 21 of square of a binomial, 368 of sum and difference of two terms, 368 of two binomials, 367 of two polynomials, 366–367 verbal phrases for, 45 by zero, 433 Multiplication Property of Equations, 75–76, 92, 150, 485, 716 Multiplication Property of Inequalities, 126, 151 Multiplication Property of One, 41, 66 Multiplication Property of Zero, 433 Multiplicative identity, 63 Multiplicative inverse, 41, 64. See also Reciprocal


M Magic square, 36 Maximum value of a quadratic function, 615–618, 622–623, 651 Mersenne prime number, 442 Minimum value of a quadratic function, 615–618, 622–623, 651 Mixture problems percent mixture, 78–80, 115–116 value mixture, 113–114 Monomial(s), 345, 357, 389 common factor, 401 degree of, 345, 389 power of, 346 products of, 345, 353 quotients of, 347–350, 353–354 simplifying powers of, 346 Motion problems, 117–118, 321, 325–326, 491, 495–496, 508. See also Uniform motion problems Multiple, least common (LCM), 19, 92–93, 461, 469, 505 Multiplication, 7–8, 65 Associative Property of, 40, 66 of binomials, 367 Commutative Property of, 41, 66

N nth root, 517, 519, 553, 554 Napier, John, 672 Natural exponential function (e x), 662, 700 Natural logarithm (ln), 670, 700 Natural number, 3, 63 Negative exponent, 348, 353–354, 390 Negative integer, 3 as an exponent, 348, 349, 390 Negative number, 126 Negative reciprocal, 274 Negative slope, 254, 286 Negative square root, 543 Newton, Isaac, 348 Nonagon, 177 Nonlinear inequalities, 589–592 Not equal to sign, 73 Null set, 55, 64 Number(s) absolute value, 4 complex, 543–548 imaginary, 543 integer, 3 irrational, 24, 527

I5

natural, 3 polygonal, 182, 283 rational, 17 real, 24, 63 Number line absolute value and, 137–138 addition on, 5 graph of an integer on, 3 multiplication on, 7 Number theory, 442 Numerical coefficient, 37, 64

O Obtuse angle, 164, 208 Obtuse triangle, 178, 209 Octagon, 177 Odd integer, 106 One in division, 9, 65 Multiplication Property of, 41, 66 One-to-one function, 639–640, 643, 651 One-to-one property of logarithms, 672, 701 Operations on decimals, 20, 21 on functions, 631–632, 652 Opposite, 4, 40, 63 Ordered pair, 219–223, 229, 285, 641 Ordered triple, 312 Order of Operations Agreement, 33–34, 66, 73, 715 and calculators, 62 Ordinate, 219, 285 Origin, 219, 285 Output, 609, 650

P Parabola, 609, 650 application problems, 616–618 axis of symmetry, 610, 650, 651 equation of, 609, 650 graphs of, 609–610, 650 intercepts of, 612, 614, 621–622, 651 vertex of, 610, 615, 650 Parallel lines, 162, 168, 207, 273 slopes of, 273–274, 276, 287 Parallelogram, 178, 208 area of, 183, 210 Parkinson, C. Northcote, 689 Parkinson’s Law, 689 Parsec, 351 Patterns, 283 Pentagon, 177 Percent(s), 18, 63 application problems, 26, 77–80 and decimals, 18, 65 and fractions, 18, 65 Percent equation, 77–80 Percent mixture problems, 78–80, 115, 151 Percent-of-light equation, 694 Perfect cube, 425, 443 Perfect powers, 521–522, 526. See also Perfect cube; Perfect square

Index

Perfect square, 24, 63, 423, 443 Perfect-square factor, 24 Perfect-square trinomial, 423–424, 571 Perimeter(s), 178–179, 181, 206, 210 Perpendicular lines, 163, 207, 274–276 slopes of, 274–275, 287 pH equation, 690 Physics, application problems, 148 Pi, 180 Plane, 161, 219, 312 Plane figure, 161 Plane geometry, 177–180 Plotting points, 219. See also Graphing; Graphs Point, 161 Point-slope formula, 264–265, 287 Polya, George, 551 Polya’s Four-Step Process, 551 Polygon(s), 177, 209 Polygonal numbers, 182, 283 Polynomial(s), 357, 389 addition of, 360–362, 364, 391 additive inverse of, 360–362, 391 binomial, 357 degree of, 357, 389 descending order, 357 division of, 375–380, 382–384 factoring, see Factoring, polynomials greatest common factor of, 401–402 least common multiple of, 461 monomial, 345, 357 multiplication of, 365–369 nonfactorable over the integers, 408, 443 prime, 386 subtraction of, 360–362, 364, 391 synthetic division, 378–380 trinomial, 357 Polynomial function(s), 357–360, 389 evaluating, 357–360, 363, 380–381 graph of, 358–360, 625 Polynomial number, 283 Positive integer, 3 Positive number, 126 Positive slope, 254, 286 Power(s) of exponential expressions, 346 of products, 346 of quotients, 348 simplifying, 22–23 verbal phrases for, 45 Prime factors, 427 Prime number, 442 Prime polynomial, 408 Principal square root, 24, 521, 553 Principle of Zero Products, 433–434, 444, 563, 599 application problems, 435–436 Problem-solving techniques, 597 by algebraic manipulation, 647 by finding a pattern, 283 by graphing techniques, 647 proof by contradiction, 697

by solving an easier problem, 333 by trial and error, 147, 205 Product(s), 7–8, 65 expressing trinomials as, see Factoring simplifying powers of, 346 See also Multiplication Product Property of Radicals, 527, 554 Product Property of Square Roots, 24–25 Proof by contradiction, 697 Properties Addition Property of Equations, 92, 150, 485, 716 Addition Property of Inequalities, 125, 151 Associative Property of Addition, 39, 66 Associative Property of Multiplication, 40, 66 Commutative Property of Addition, 39, 66 Commutative Property of Multiplication, 41, 66 Distributive Property, 38, 66 Inverse Property of Addition, 40, 66 Inverse Property of Logarithms, 671–673, 684, 685, 701 Inverse Property of Multiplication, 41, 66 Multiplication Property of Equations, 75–76, 92, 150, 485, 716 Multiplication Property of Inequalities, 126, 151 Multiplication Property of One, 41, 66 Multiplication Property of Zero, 433 One-to-One Property of Exponential Functions, 669 One-to-One Property of Logarithms, 672, 701 Product Property of Radicals, 527, 554 Product Property of Square Roots, 24–25 Property of Raising Both Sides of an Equation to a Power, 537, 554 Property of the Composition of Inverse Functions, 642, 652 Quotient Property of Radicals, 531, 554 Proportion, 477, 506 application problems, 478–480 Proportionality, constant of, 497, 506 Protractor, 163 Ptolemy, 3 Pyramid regular, 195 surface area of, 199 volume of, 196 Pythagoras, 539 Pythagorean Theorem, 539, 554 Pythagorean triple, 537

Q Quadrant, 219, 285 Quadratic equation(s), 433, 443, 563, 598 applications of, 593–594 degree, 563 discriminant of, 579, 599, 613–614, 651 rational, reducible to, 584–588 solving by completing the square, 571–576, 600 solving by factoring, 433–434, 437–438, 563–564, 567–568, 599 solving by quadratic formula, 577–582 solving by taking square roots, 565–566, 569–570, 600 solving by using a graphing calculator, 597–598 standard form, 433, 443, 563, 598 writing, given solutions, 564–565, 568–569, 599 Quadratic formula, 577–579, 581–582, 600 Quadratic function, 357, 389, 650 graph of, 609–611, 619–620, 651 maximum and minimum of, 615, 622–623, 651 Quadratic inequality, 589–592, 599 Quadratic in form, 426–427, 431, 443, 583–584, 587, 599 Quadrilateral, 178–179, 208 Quotient(s), 8 power of, 348 Quotient Property of Radicals, 531, 554

R Radical equation, 537, 553 reducible to quadratic, 584–588 Radical expression(s) addition of, 528–529, 533–534 application problems, 539–540 division of, 531–532, 535–536 equations containing, 537–538 exponential form of, 519–520, 525, 554 index of, 499 multiplication of, 529–530, 534–535 rationalizing denominators of, 531–532, 553 simplest form of, 527–531 simplifying, 24–25, 521–522, 526, 527–528, 533 subtraction of, 528–529, 533–534 Radical function, 626 Radical sign, 63, 519, 553 Radicand, 24, 63 Radius of a circle, 180, 209 of a sphere, 195 Range, 230, 233, 286, 641 estimating from a graph, 627 of exponential function, 662


I6


Index

of parabola, 609, 611 of quadratic function, 595, 596 Rate, 477, 505 interest, 78, 150, 303, 336 motion problems, 117 in proportions, 477 in work problems, 489 Rate-of-wind or current problems, 321–322, 491–492, 594 Ratio, 477, 505 Rational equation, reducible to quadratic, 588 Rational exponents, 517–519, 553 Rational expression(s), 453, 505 addition of, 461, 463–464, 466–468, 507 division of, 456, 459–460, 507 expressing in terms of the LCD, 440 multiplication of, 454–455, 458–459, 506 reciprocal of, 456, 505 rewriting in terms of a common denominator, 461–464 simplifying, 453–454, 457, 506 solving equations containing, 491, 508 subtraction of, 461, 463–464, 466–468, 507 Rationalizing a denominator, 531–532, 553 Rational number(s), 17 addition of, 19–20 applications of, 26 decimal notation for, 17 division of, 21 as exponents, 517–518 multiplication of, 20, 21 subtraction of, 19–20 Ray, 161, 208 Real number, 24, 63 Reciprocal(s), 21, 41, 64, 456 and division, 21 negative, 264 of rational expressions, 456, 505 Rectangle, 178, 208 area of, 182, 210 perimeter of, 179, 210 Rectangular coordinate system, 219–222, 285 Rectangular solid, 195, 209 surface area of, 198, 199, 210 volume of, 196–197, 210 Regular polygon, 177, 209 Regular square pyramid, volume, 196, 210 Relation, 229, 286 Remainder Theorem, 381, 391 Repeating decimal, 17, 24, 63 Rhombus, 178 Richter, Charles F., 691 Richter scale, 691 Richter scale equation, 695 Right angle, 163, 208 Right circular cone, 195, 209

surface area of, 199, 210 volume of, 196 Right circular cylinder, 195 surface area of, 199, 210 volume of, 196 Right square pyramid, 195, 209 surface area of, 199, 210 Right triangle, 178, 209, 539 Root(s) cube, 403, 425, 522 double, 563 nth, 517, 519, 553, 554 of perfect powers, 521, 522 square, 24, 401, 522 Roster method, 55, 64 Rudolff, Christoff, 519 Rule for Dividing Exponential Expressions, 349, 351 Rule for Multiplying Exponential Expressions, 345, 346 Rule for Negative Exponents on Fractional Expressions, 349, 351 Rule for Simplifying Powers of Exponential Expressions, 346, 683 Rule for Simplifying Powers of Products, 346 Rule for Simplifying Powers of Quotients, 348, 351 Rules of Exponents, 349–351

S Scalene triangle, 177, 209 Scatter diagrams, 223–224, 286 Scientific notation, 351–352, 388, 390 Second coordinate, see Ordinate Second-degree equation, 563, 598. See also Quadratic equation Seismology, application problems, 695–696 Sequence, 61 Set(s), 3, 55, 64 element of, 55, 64 empty, 55 graphs of, see Graph(s) intersection of, 55–56, 58, 64 null, 55 solution, 125 union of, 55–56, 64 writing by using roster method, 55–56, 64 writing by using set-builder notation, 56, 64 Set-builder notation, 56–57, 64 Sides of an angle, 162 of a polygon, 177, 209 Similar triangles, 478–480, 508 Simple interest, 78, 150, 303, 336 Simplest form of an exponential expression, 346 of a radical expression, 24, 527–531 of a rate, 477 of a ratio, 477

I7

of a rational expression, 453, 457, 506 Simplifying complex fractions, 469 complex numbers, 543–544 exponential expressions, 23, 346, 390 expressions with rational exponents, 517–519 numerical expressions, 33–34 powers of exponential expressions, 346, 390 powers of products, 346, 390 powers of quotients, 348, 390 radical expressions, 24–25, 521–522, 526, 527–528, 533 rational expressions, 453–454, 457, 506 variable expressions, 38–44, 716 Slant height, 195 of a cone, 195 of a pyramid, 195 Slope, 253–254, 256, 286 Slope formula, 253, 287 Slope-intercept form of linear equation, 257–258, 264, 287 Slope of a line, 253–255, 287 applications of, 255, 259 of parallel lines, 273–274, 287 of perpendicular lines, 274–275, 287 Solids, 195–199 Solution(s) of equations, see Equation(s), solution of; Solving equations extraneous, 538 of inequalities, see Solution set of an inequality of linear equation in three variables, 314 of linear equation in two variables, 221, 285 of quadratic equation, see Quadratic equation(s) of a system of equations, 297, 312–313, 335 Solution set of an inequality, 125, 149 linear, 279 quadratic, 589, 592 system, 329 Solving equations, 73, 92–99, 149 with absolute value, 143–144 containing parentheses, 97 containing radical expressions, 525–526, 537–538, 544, 585 exponential, 683–684, 701 by factoring, 433–434, 437–438, 563 first-degree, 92–99 fractional, 473–476, 508, 588 literal, 485 logarithmic, 685–686, 698 proportions, 477 quadratic, see Quadratic equation(s) quadratic in form, 583–584, 587 rational, 455, 457 reducible to quadratic, 584–588

Index

Solving equations (continued) using the Addition Property, 74–75 using the Multiplication Property, 76 Solving inequalities with absolute value, 144–145 applications of, 118, 128 compound, 128, 132 in one variable, 125 systems of, 329 using the Addition Property, 125–126 using the Multiplication Property, 126–127 Solving proportions, 477 Solving systems of equations, see Systems of equations, solving Sørenson, Søren, 690 Sphere, 195, 209 center of, 195 diameter of, 195 radius of, 195 surface area of, 199, 210 volume of, 196–197, 210 Square, 178, 208 area of, 180, 183, 210 of a binomial, 368, 571 of a number, 22 perfect trinomial, 423–424 perimeter of, 179, 210 of a variable, 45 Square function, 230, 231 Square root(s), 24, 63, 423 approximating, 24 of negative numbers, 543 of a perfect square, 24, 423 principal, 24, 63 product property of, 24–25, 66 simplifying numerical, 25 Standard form of a quadratic equation, 433, 443, 563, 598 Stevin, Simon, 17, 21 Straight angle, 164, 203 Substitution method for solving systems of equations, 300–301 Subtraction, 65 as addition of opposites, 6 of complex numbers, 544–545, 549, 554 of decimals, 20 of fractions, 19–20, 66 of functions, 631 of integers, 6 of polynomials, 360–362, 364, 391 of radical expressions, 528–529, 533–534 of rational expressions, 461, 463, 466–468, 507 of rational numbers, 19–20 verbal phrases for, 45 Sum of additive inverses, 40 Sum or difference of two cubes, 425 Supplementary angles, 164, 165, 166, 208 Surface area, 198 of geometric solids, 199

Symbols angle, 162 approximately equal to, 21 empty set, 55 equal to, 74 greater than, 3 greater than or equal to, 3 imaginary number (i), 543 inequality, 3 intersection, 55 less than, 3 less than or equal to, 3 not equal to, 9 parallel, 162 perpendicular, 163 pi, 180 square root, 24 union, 55 Symmetry, 207 axis of, see Axis of symmetry Synthetic division, 378–381, 383–385, 391 and evaluating polynomials, 383–385 Systems of equations, 297–300, 335 applications of, 321–324 dependent, 298, 310, 313, 335 graphing, 297–300 inconsistent, 298, 313, 335 independent, 297, 313, 335 solution of, 297, 312–313, 335 solving by the addition method, 309–311, 312–315, 336 solving by graphing, 297–299, 305–306, 336 solving by using a graphing calculator, 333–335 solving by the substitution method, 300–302, 307, 336 in three variables, 312–316 in two variables, 297–301, 335 Systems of inequalities, 329–332, 336

T Tables, creating, 441 Terminating decimal, 17, 63 Term(s), 37, 64, 716 coefficient of, 37 combining like, 38–39 constant, see Constant term like, 38 of a quotient, 378–379 of a sequence, 61 variable, 37, 716 Theorem(s) Pythagorean, 539, 554 Remainder, 381, 391 Tolerance, 149 of a component, 141 Translating sentences into equations, 106–109 Translating verbal expressions into variable expressions, 44–46, 106–109 Transversal, 167, 208

Trapezoid, 178 area of, 184–186, 210 isosceles, 178 Trial-and-error method, 147, 205–206, 668 Trial factors, 415–416, 419–420 Triangle(s), 169, 177, 209 acute, 178, 209 angles of, 169–170, 209, 479 application problems, 478–480, 540 area of, 184, 210 equilateral, 177, 209 isosceles, 177, 209 obtuse, 178, 209 perimeter of, 178, 210 right, 178, 209, 539 scalene, 177, 209 similar, 478–480, 508 Trinomial, 357, 389 factoring, 407–424 perfect-square, 423–424, 571 quadratic in form, 426, 431, 443, 599

U Undefined slope, 254, 256 Uniform motion equation, 117, 151 Uniform motion problems, 81–83, 117, 321, 325–326, 491–492, 495–496, 508 Union of sets, 55, 64 Upper limit, 141, 149

V Value of a function, 232, 286 Value mixture problems, 113–114, 150 Variable, 3, 37, 64, 92, 95, 106, 149, 715 dependent, 230, 286 independent, 230, 286 square of, 45 value of, 715 Variable expression(s), 37, 64, 715 evaluating, 37, 715 like terms of, 38–39 simplifying, 38–43, 716 translating into, from verbal expressions, 44–46 value of, 715 Variable part of a variable term, 37, 64 Variable term, 37, 64 Variation, 497–501, 506 Verbal expressions, translating into variable expressions, 44–46 Vertex, 162 of an angle, 162, 208 of a cone, 195 of a parabola, 610, 615, 650, 651 Vertical angles, 166, 170, 208, 479 Vertical line, 228 Vertical-line test, 626, 652, 679 Volume, 196 formulas for, 210 of geometric solids, 196


I8

Index

W

Y

Water displacement, 148 Work problems, 489–490, 493–494, 508, 510, 514, 593

y-axis, 221 y-coordinate, 221, 285 y-intercept(s), 246, 286 of a line, 246–247 of a parabola, 612

X x-axis, 221, 625 x-coordinate, 221, 285 x-intercept(s), 246, 286, 612, 650 discriminant and, 613–614 of a line, 246–247 of a parabola, 612, 614, 651 xy coordinate system, 221 xyz coordinate system, 312

Z z-axis, 312 Zero(s) absolute value of, 4 Addition Property of, 39 in the denominator of a rational expression, 9, 65, 349, 453

I9

division by, 9, 65, 349, 453 as an exponent, 347, 390 of a function, 612–614, 621–622, 650 Multiplication Property of, 433 origin of symbol, 3 Principle of Zero Products, 433, 444, 563 Zero-level earthquake, 691 Zero slope, 254, 286

Index of Applications (Continued from inside front cover)


O Oceanography, 560 Oil spills, 637 The Olympics, 623 P Packaging, 204 Paint, 204, 212 Parachuting, 240 Parks, 193, 212 Patios, 192, 212 Patterns in mathematics, 283, 441 Paving, 387 Pendulums, 540, 542 Percent mixture problems, 78, 80, 89, 90, 115, 116, 120–122, 154, 156, 158, 342, 514, 658 Perfume, 78 Periscopes, 540 Petroleum, 202 Pharmacology, 90 Photography, 446 Physics, 99, 104, 278, 352, 356, 387, 388, 394, 436, 439, 497, 499, 500, 501, 502, 540, 541, 558, 595, 596, 617, 623, 658, 667, 688, 708, 712 Pizza, 214 Planets, 53 Playgrounds, 186, 189 Postage, 695 Printers, 46 Psychology, 240 Publishing, 440 Pulleys, 54 Purchasing, 322–323, 324, 326, 327, 340, 342 Q Quilting, 189

R Radioactivity, 693, 704, 706 Ranching, 624 Rate-of-wind problems, 322, 325, 326, 342, 491, 496, 596, 708 Record sales, 88 Real estate, 237, 387 Rocketry, 482 Rockets, 595 Roller coasters, 251 Running, 255 S Safety, 595, 624 Sailing, 53 Sales tax, 256 Satellites, 542 School enrollment, 88 Seismology, 691, 692, 695, 696, 704 Sewing, 190 Shipping, 236, 237 Solar system, 388 Sound, 695, 704 Sports, 88, 112, 181, 260, 364, 439, 446, 484, 576, 582, 602, 604, 624, 695 Sports equipment, 53 Sports leagues, 333 The stock market, 32, 712 Storage units, 201 Sum of natural numbers, 439 Summer camp, 292 Surveys, 481 Swimming pools, 46 Symmetry, 207 T Tanks, 658 Taxes, 482 Technical support, 111

Telecommunications, 111, 130, 134, 135, 251, 271 Telescopes, 192, 642 Television, 542 Temperature, 10, 70, 135, 240, 260, 267 Testing, 15, 68, 130 Ticket sales, 327, 338 Time, 396 Tolerance, 141, 142, 145, 146, 606 Tornadoes, 53 Transportation, 124, 496 Travel, 53, 124, 259, 290, 294, 342, 398, 450, 452, 560, 712 Typing, 682, 692 U Uniform motion problems, 81–83, 90, 91, 117, 118, 122–124, 153, 154, 156, 158, 321, 322, 325, 326, 342, 491, 492, 495, 496, 594, 596, 602, 604, 708, 711, 712 Union dues, 111 Utilities, 228 V Value mixture problems, 113, 114, 119, 120, 154, 156, 158, 294, 398, 658, 708 W Water-current problems, 321, 322, 325, 496, 594, 596, 602, 604, 712 Water displacement, 148 Wildland fires, 224 Word game, 551 Work problems, 489, 490, 493, 494, 496, 593, 596, 602 World population, 53

x ENTER ENTER x 22 2ENTER ENTER x 2 x ENTER 2

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d/c Change decimal to fraction

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Photo courtesy of Casio, Inc.

Power of a number (See Note 1 below.)

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Enter a negative number (See Note 2 below.)

2 5

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Operations with a b/c 11 parentheses a b/c b/c b/c a a b b a b/c ENTER aa b/c/c a c / 25 2nd % ENTER a b/c a b2nd c /2nd ENTER ENTER Operations with 2nd percent 2.75

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Change decimal to fraction or fraction to decimal

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x NOTE 1: Some calculators use the y key to calculate a power. For those calculators, enter 13 yx 4 to evaluate 134.

FD ENTER

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a b/c b aabb//cc a /c a b aab/bc/c a b/c 11 /c b b b b a a /c a /c a /c /c ENTER aabb//cc aabb//cc 2nd aabb//cc 11*25% ENTER 2nd ENTER b/c ENTER a b/c a a b2nd c /2nd ENTER

( ) ENTER ( ) of a number 6 ( ) ( ENTER a b/c a b/c a b/c b/c 3 a b a b/c b/cc 4 6 a b/c 2 a b/c 3 a a b/c a b/c) a b/c ENTER ( ENTER Operations 2nd 6 2 33 4 ( )ENTERENTER 2nd on fractions 7 5 12 2 3 5 6 7 3 4 12

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a b/c a b/c 28561 b/c a a bb/c ) ENTER a /c a b c ENTER 36 a b/c )/ ENTER ENTER 2nd ) ENTER () (36) 6

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π ENTER The value of π ENTER 3.141592654 ( )ENTER ( ENTER ) ENTERENTER ENTER 2 Power of ( ( 13 4 ) ) ENTER x

root2nd Square of a number

A b/c

Operations ENTER 6 2 33 4 ENTER 2nd 2nd ENTER on fractions ENTER ENTER 2nd 2nd 7 5/12 2 3 5 6 7 3 4 12

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2.75

Operations with percent

Used to complete an operation SHIFT

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3.141592654

The value of π

Index of Applications A Advertising, 88 Aeronautics, 338, 340 Agriculture, 53, 202, 214, 255 Airports, 237 Animal science, 251 Annual earnings, 216 Appliances, 145, 202 Aquariums, 202, 204, 387 Art, 328, 482 Astronomy, 355, 356, 388, 394, 560, 682 Athletic fields, 192 Athletics, 240 The atmosphere, 16 Automobile rebates, 638 Automobiles, 142, 146, 154 Automotive technology, 124, 240, 556 Aviation, 270 B Balance of trade, 70 Ballooning, 204 Banking, 135, 136, 708 Banners, 189 Baseball, 70, 510 Baseball cards, 68 Boating, 338 Boiling points, 271 Business, 26, 31, 104, 105, 228, 237, 267, 171, 326, 327, 482, 488, 500, 501, 623, 647, 648, 649 C Calories, 270, 271 Candy, 68 Carbon dating, 690 Carpentry, 72, 109, 112, 181, 190, 214, 450, 616, 708 Carpeting, 189, 193, 387 Cartography, 481 Cereal, 224 Chemistry, 15, 54, 88, 90, 124, 228, 327, 623, 690, 692, 694, 695 Clubs, 484 Cocoa, 77 Coins, 328 Colors, 80 Compensation, 135, 154, 251, 271, 342, 501, 648 Compound interest, 689, 692, 693, 704, 708, 712 Computers, 72, 146, 352, 602 Computer science, 111 Consecutive integer problems, 107, 108, 110, 130, 136, 435, 472, 602 Conservation, 482 Construction, 111, 194, 260, 271, 290, 500, 596, 623

Consumerism, 135, 156 Cooking, 481 Cost of labor, 248 Credit ratings, 699–700 Cycles, 190, 214 D Demography, 694 Depreciation, 256, 292, 294, 606 Drag racing, 387 Dye mixtures, 89 E Earth science, 190, 694 Education, 88, 112, 136, 154, 711 Electric vehicles, 36 Electronics, 146, 342, 500, 502, 510, 512 Energy, 112, 239, 556 Entertainment, 108 F Fabric mixtures, 89 The Federal budget, 32 The Federal government, 356 Fencing, 189 Finances, 326, 335 Fire science, 624, 633 First-class postage rates, 646 Fish hatcheries, 201 Flooring, 190, 193 Flour mixtures, 326 Food mixtures, 80, 90 Football, 79 Fountains, 624 Framing, 190, 212 Fuel consumption, 260, 271 G Gardening, 440, 446, 450 Gardens, 189, 192 Geography, 15, 595 Geology, 356 Geometry, Chapter 3; see also 111, 112, 130, 134, 278, 328, 369, 370, 373, 374, 386, 394, 396, 398, 406, 436, 439, 440, 448, 450, 460, 479, 480, 482–484, 498, 501, 539, 542, 594, 595, 596, 602, 606, 618, 654, 656, 712 Government, 88 Grading scale, 646 H Halloween spending, 32 Health science, 327 Heart rate, 248 Home maintenance, 193, 260, 556, 606 Horse racing, 387 The hospitality industry, 289

Human proportions, 248 I Ice cream, 596 Image projection, 446 Income, 88 Indirect measurement, 484 Inductive reasoning, 61 Insurance, 514 Integer problems, 107, 108, 110, 130, 134, 136, 154, 156, 158, 398, 435, 442, 472, 602 Interior decorating, 108, 186, 192, 193, 194, 478, 512 The Internal Revenue Service, 53 Internet service, 326 Investment problems, 303, 304, 307, 308, 324, 334, 335 Investments, 46, 78, 80, 89, 112, 216, 327, 328, 338, 340, 398, 450, 560, 696, 704, 708, 712 Irrigation, 192 J Juice mixtures, 89 L Landscaping, 193, 212, 440, 512 Lever systems, 99, 104 Life expectancy, 289 Light, 502, 505, 691, 694, 712 Loan payments, 478 Logic, 503 Lottery tickets, 484 M Magnetism, 502 Manufacturing, 252, 290, 326, 327, 637, 638 Marathons, 224 Mathematics, 588, 624 Measurement, 356, 387 Mechanics, 142, 145, 154, 156, 502, 606 Media, 259 Medication, 499 Metallurgy, 89 Metalwork, 54, 109 Meteorology, 16, 260 Mining, 617 Mixture problems, 78, 80, 89, 90, 113–116, 119–122, 154, 156, 158, 294, 342, 398, 514, 658, 708 Money, 68 Music, 46, 79, 658 N Natural resources, 53, 695 Number problems, 46, 51, 52, 68, 70, 618 Number sense, 406, 438, 448 (Continued on page I9 of the Subject Index)

TI-83 Plus/84 Plus* WINDOW Xmin = –10 Xmax = 10 Xscl = 1 Ymin = –10 Ymax = 10 Yscl = 1 Xres = 1

2nd

2nd

TABLE

X

2nd

TBLSET

TABLE SETUP TblStart=0 ∆Tbl=1 Indpnt: Auto Depend: Auto

STAT PLOT

2nd

CALC

CALCULATE 1 : value 2: zero 3: minimum 4: maximum 5: intersect 6: dy/dx 7: ∫ f(x)dx

Ask Ask

STAT PLOTS 1 :Plot1…On L1 L2 2:Plot2…Off L1 L2 3:Plot3…Off L1 L2 4 PlotsOff

Y1

0 1 2 3 4 5 6

–3 –1 1 3 5 7 9

FUNCTION 1 : Y1 2: Y 2 3: Y 3 4: Y 4 5: Y 5 6: Y 6 7 Y7

X=0

ZOOM MEMORY 1 : ZBox 2: Zoom In 3: Zoom Out 4: ZDecimal 5: ZSquare 6: ZStandard 7 ZTrig

VARS Y-VARS 1 : Function... 2: Parametric... 3: Polar... 4: On/Off...

Plot1 Plot2 Plot3 \Y 1 = \Y 2 = \Y 3 = \Y 4 = \Y 5 = \Y 6 = \Y 7 =

2nd NAMES 1 : [A] 2 : [B] 3 : [C] 4 : [D] 5 : [E] 6 : [F] 7 [G]

VARS Y-VARS 1 : Window... 2: Zoom... 3: GDB... 4: Picture... 5: Statistics... 6: Table... 7: String...

MATRX

Normal Sci Eng Float 0123456789 Radian Degree Func Par Pol Seq Connected Dot Sequential Simul Real a+bi re^θi Full Horiz G–T

MATH EDIT 3x4 3x4 3x4

Access functions and menus written in gold

Move the cursor on the graphics screen or scroll through a menu

Enter the variable x MATH NUM CPX PRB 1 : abs( 2 : round( 3 : iPart( 4 : fPart( 5 : int( 6 : min( 7 max(

Enter an exponent

MATH NUM CPX PRB 1 : conj( 2: real( 3: imag( 4: angle( 5: abs( 6: Rect 7: Polar

2nd ALPHA

MATH NUM CPX PRB 1 : Frac 2 : Dec 3:3 4 : 3 √( 5 : x√ 6 : fMin( 7 fMax(

ENTRY - Recall last calculation SOLVE - Used to solve some equations and to solve for a variable in financial calculations Used to complete an operation

2nd

ANS - Recall last answer Used to enter a negative number

- Reciprocal of last entry EDIT CALC TESTS 1 : 1–Var Stats 2 : 2–Var Stats 3 : Med-Med 4 : LinReg(ax+b) 5 : QuadReg 6 : CubicReg 7 QuartReg

x2 LOG

LN EDIT CALC TESTS 1 : Edit… 2 : SortA( 3 : SortD( 4 : ClrList 5 : SetUpEditor

STO

- Square last entry;

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: Square root of next entry

- Common logarithm (base 10); - Natural logarithm (base e); - Store a number;

2nd

2nd

2nd

10x : 10 to the x power, antilogarithm of x

ex : Calculate a power of e

RCL : Recall a stored variable

*The calculator shown is a TI-83 Plus. The operations for a TI-84 Plus are exactly the same. Photo courtesy of Texas Instruments Incorporated

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